Calculus: Early Transcendentals
12th Edition
ISBN: 9781119778189
Author: Anton, Howard, Bivens, Irl C., Davis, Stephen
Publisher: WILEY
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Chapter 6.9, Problem 22ES
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3. (i) Let A, B, C, D, E be positive constants. Find the limit
n³ D
lim A(BC)³ +1 en 4+E.
n→∞
[6 Marks]
(ii) Let X be a random variable with the standard normal distribution,
i.e. X has the probability density function
Let
1
f(x)
√2π
n3x2
Xn = 100(50 +X6) en³4, n = 1, 2,...,
3+1
(a) Find the limit lim∞ Xn, verify your answer.
Y = 100(50 +X) 22.
[6 Marks]
(b) Is Xn≤Y? Verify your answer.
[6 Marks]
(c) Is EY∞o? Verify your answer.
[6 Marks]
(d) Does lim∞ EXn
(e) Find lim∞ EXn.
Elim Xn? Why?
[6 Marks]
[3 Marks]
2.
Let (, F, P) be a probability space. Let X, Y be two independent ran-
dom variables such that
1
3
P(X = 0)
P(X = 1)
P(X = 2)
P(X = 3)
P(X = 4)
10
10
5'
3
P(Y = 0)
P(Y = 1)
10
1
10
7
1
P(Y2) =
P(Y = 3)
,
P(Y = 4)
10
20
20
(i) Find the probability generation function Gx(s) and Gy(s) of X and
Y, respectively.
[12 Marks]
(ii) Using Gx(s) and Gy(s), find the mean and variance of X and Y.
[12 Marks]
(iii) Find the probability generating function Gx+y(s) of X + Y.
[8 Marks]
3. (i) Using the definition of the line integral of a vector field, calculate the
line integral
L³
F.dy
of the vector field F: R² → R² given by
F(x, y) = (y, x),
and where the curve & is the unit semi-circle centred at the origin, located in
the upper half-plane and oriented in the anticlockwise direction.
Hint. Represent the curve y as the join of two curves y = 71 + 1/2 (see Example 8.9
in the Notes).
[20 Marks]
(ii) Calculate the same integral using Green's Theorem.
[10 Marks]
Chapter 6 Solutions
Calculus: Early Transcendentals
Ch. 6.1 - An integral expression for the area of the region...Ch. 6.1 - An integral expression for the area of the...Ch. 6.1 - (a) The points of intersection for the circle...Ch. 6.1 - Prob. 4QCECh. 6.1 - Find the area of the shaded region.Ch. 6.1 - Find the area of the shaded region.Ch. 6.1 - Find the area of the shaded region.Ch. 6.1 - Find the area of the shaded region.Ch. 6.1 - Find the area of the shaded region by (a)...Ch. 6.1 - Find the area of the shaded region by (a)...
Ch. 6.1 - Sketch the region enclosed by the curves and find...Ch. 6.1 - Sketch the region enclosed by the curves and find...Ch. 6.1 - Sketch the region enclosed by the curves and find...Ch. 6.1 - Sketch the region enclosed by the curves and find...Ch. 6.1 - Sketch the region enclosed by the curves and find...Ch. 6.1 - Sketch the region enclosed by the curves and find...Ch. 6.1 - Sketch the region enclosed by the curves and find...Ch. 6.1 - Sketch the region enclosed by the curves and find...Ch. 6.1 - Sketch the region enclosed by the curves and find...Ch. 6.1 - Sketch the region enclosed by the curves and find...Ch. 6.1 - Sketch the region enclosed by the curves and find...Ch. 6.1 - Prob. 19ESCh. 6.1 - Use a graphing utility, where helpful, to find the...Ch. 6.1 - Prob. 22ESCh. 6.1 - Use a graphing utility, where helpful, to find the...Ch. 6.1 - Use a graphing utility, where helpful, to find the...Ch. 6.1 - Use a graphing utility, where helpful, to find the...Ch. 6.1 - Use a graphing utility, where helpful, to find the...Ch. 6.1 - Determine whether the statement is true or false....Ch. 6.1 - Determine whether the statement is true or false....Ch. 6.1 - Determine whether the statement is true or false....Ch. 6.1 - Determine whether the statement is true or false....Ch. 6.1 - Estimate the value of k0k1 so that the region...Ch. 6.1 - Estimate the area of the region in the first...Ch. 6.1 - Find a horizontal line y = k that divides the area...Ch. 6.1 - Find a vertical line x=k that divides the area...Ch. 6.1 - (a) Find the area of the region enclosed by the...Ch. 6.1 - Find the area between the curve y=sinx and the...Ch. 6.1 - Use Newton’s Method (Section 4.7 ), where...Ch. 6.1 - Use Newton’s Method (Section 4.7 ), where...Ch. 6.1 - Use Newton’s Method (Section 4.7 ), where...Ch. 6.1 - Find the area of the region that is enclosed by...Ch. 6.1 - Referring to the accompanying figure, use a CAS to...Ch. 6.1 - Two racers in adjacent lanes move with velocity...Ch. 6.1 - The accompanying figure shows acceleration versus...Ch. 6.1 - The accompanying figure shows the rate at which...Ch. 6.1 - Find the area of the region enclosed between the...Ch. 6.1 - Show that the area of the ellipse in the...Ch. 6.1 - Suppose that f and g are continuous on a,b but...Ch. 6.2 - A solid S extends along the x-axis from x=1 to x=3...Ch. 6.2 - A solid S is generated by revolving the region...Ch. 6.2 - A solid S is generated by revolving the region...Ch. 6.2 - A solid S is generated by revolving the region...Ch. 6.2 - Find the volume of the solid that results when the...Ch. 6.2 - Find the volume of the solid that results when the...Ch. 6.2 - Find the volume of the solid that results when the...Ch. 6.2 - Find the volume of the solid that results when the...Ch. 6.2 - Find the volume of the solid that results when the...Ch. 6.2 - Find the volume of the solid that results when the...Ch. 6.2 - Find the volume of the solid that results when the...Ch. 6.2 - Find the volume of the solid that results when the...Ch. 6.2 - Find the volume of the solid that results when the...Ch. 6.2 - Find the volume of the solid that results when the...Ch. 6.2 - Find the volume of the solid that results when the...Ch. 6.2 - Find the volume of the solid that results when the...Ch. 6.2 - Find the volume of the solid that results when the...Ch. 6.2 - Prob. 16ESCh. 6.2 - Find the volume of the solid that results when the...Ch. 6.2 - Find the volume of the solid whose base is the...Ch. 6.2 - Find the volume of the solid that results when the...Ch. 6.2 - Find the volume of the solid that results when the...Ch. 6.2 - Find the volume of the solid that results when the...Ch. 6.2 - Find the volume of the solid that results when the...Ch. 6.2 - Find the volume of the solid that results when the...Ch. 6.2 - True-False Determine whether the statement is true...Ch. 6.2 - True-False Determine whether the statement is true...Ch. 6.2 - True-False Determine whether the statement is true...Ch. 6.2 - True-False Determine whether the statement is true...Ch. 6.2 - Find the volume of the solid that results when the...Ch. 6.2 - Let V be the volume of the solid that results when...Ch. 6.2 - Find the volume of the solid generated when the...Ch. 6.2 - Find the volume of the solid generated when the...Ch. 6.2 - Consider the solid generated by revolving the...Ch. 6.2 - Consider the solid generated by revolving the...Ch. 6.2 - Find the volume of the solid that results when the...Ch. 6.2 - Find the volume of the solid that results when the...Ch. 6.2 - Find the volume of the solid that results when the...Ch. 6.2 - Find the volume of the solid that results when the...Ch. 6.2 - A nose cone for a space reentry vehicle is...Ch. 6.2 - Find the volume of the solid whose base is the...Ch. 6.2 - In parts (a)-(c) find the volume of the solid...Ch. 6.2 - Prob. 50ESCh. 6.2 - Use a CAS to estimate the volume of the solid that...Ch. 6.2 - Use a CAS to estimate the volume of the solid that...Ch. 6.2 - Use a CAS to estimate the volume of the solid that...Ch. 6.2 - Use a CAS to estimate the volume of the solid that...Ch. 6.2 - The accompanying figure shows a spherical cap of...Ch. 6.2 - If fluid enters a hemispherical bowl with a radius...Ch. 6.2 - The accompanying figure shows the dimensions of a...Ch. 6.2 - Use the result in Exercise 55 to find the volume...Ch. 6.2 - As shown in the accompanying figure, a cocktail...Ch. 6.2 - Find the volume of the torus that results when the...Ch. 6.2 - A wedge is cut from a right circular cylinder of...Ch. 6.2 - Find the volume of the wedge described in Exercise...Ch. 6.2 - Two right circular cylinders of radius r have axes...Ch. 6.2 - In 1635 Bonaventura Cavalieri, a student of...Ch. 6.3 - Let R be the region between the x-axis and the...Ch. 6.3 - Let R be the region described in Quick Check...Ch. 6.3 - A solid S is generated by revolving the region...Ch. 6.3 - Use cylindrical shells to find the volume of the...Ch. 6.3 - Use cylindrical shells to find the volume of the...Ch. 6.3 - Use cylindrical shells to find the volume of the...Ch. 6.3 - Use cylindrical shells to find the volume of the...Ch. 6.3 - Use cylindrical shells to find the volume of the...Ch. 6.3 - Use cylindrical shells to find the volume of the...Ch. 6.3 - Use cylindrical shells to find the volume of the...Ch. 6.3 - Use cylindrical shells to find the volume of the...Ch. 6.3 - Use cylindrical shells to find the volume of the...Ch. 6.3 - Use cylindrical shells to find the volume of the...Ch. 6.3 - Use cylindrical shells to find the volume of the...Ch. 6.3 - Use cylindrical shells to find the volume of the...Ch. 6.3 - Use cylindrical shells to find the volume of the...Ch. 6.3 - Use cylindrical shells to find the volume of the...Ch. 6.3 - Use cylindrical shells to find the volume of the...Ch. 6.3 - True-False Determine whether the statement is true...Ch. 6.3 - True-False Determine whether the statement is true...Ch. 6.3 - True-False Determine whether the statement is true...Ch. 6.3 - True-False Determine whether the statement is true...Ch. 6.3 - Use a CAS to find the volume of the solid...Ch. 6.3 - Use a CAS to find the volume of the solid...Ch. 6.3 - Consider the region to the right of the y-axis ,...Ch. 6.3 - Let K1 and R2 be regions of the form shown in the...Ch. 6.3 - (a) Use cylindrical shells to find the volume of...Ch. 6.3 - Let f be continuous and nonnegative on a,b , and...Ch. 6.3 - Using the method of cylindrical shells, set up but...Ch. 6.3 - Use cylindrical shells to find the volume of the...Ch. 6.3 - Use cylindrical shells to find the volume of the...Ch. 6.3 - Use cylindrical shells to find the volume of the...Ch. 6.3 - As shown in the accompanying figure, a cylindrical...Ch. 6.3 - Use cylindrical shells to find the volume of the...Ch. 6.3 - Let Vx and Vy be the volumes of the solids that...Ch. 6.3 - (a) Find the volume V of the solid generated when...Ch. 6.4 - A function f is smooth on a,b if f is on a,b .Ch. 6.4 - If a function f is smooth on a,b , then the length...Ch. 6.4 - The distance between points 1,0 and e,1 is .Ch. 6.4 - Let L be the length of the curve y=lnx from 1,0 to...Ch. 6.4 - Use the Theorem of Pythagoras to find the length...Ch. 6.4 - Find the exact arc length of the curve over the...Ch. 6.4 - Find the exact arc length of the curve over the...Ch. 6.4 - Find the exact arc length of the curve over the...Ch. 6.4 - Find the exact arc length of the curve over the...Ch. 6.4 - Find the exact arc length of the curve over the...Ch. 6.4 - True-False Determine whether the statement is true...Ch. 6.4 - True-False Determine whether the statement is true...Ch. 6.4 - True-False Determine whether the statement is true...Ch. 6.4 - True-False Determine whether the statement is true...Ch. 6.4 - Express the exact arc length of the curve over the...Ch. 6.4 - Express the exact arc length of the curve over the...Ch. 6.4 - Consider the curve y=x2/3 . (a) Sketch the portion...Ch. 6.4 - Consider the curve segments y=x2 from x=12 to x=2...Ch. 6.4 - Follow the directions of Exercise 17 for the curve...Ch. 6.4 - Follow the directions of Exercise 17 for the curve...Ch. 6.4 - Let y=fx be a smooth curve on the closed interval...Ch. 6.4 - Use the result of Exercise 20 to show that the arc...Ch. 6.4 - A basketball player makes a successful shot from...Ch. 6.4 - Prob. 23ESCh. 6.4 - As shown in the accompanying figure, a horizontal...Ch. 6.4 - These exercises assume familiarity with the basic...Ch. 6.4 - Use the arc length formula from Exercise 26 to...Ch. 6.4 - Use the arc length formula from Exercise 26 to...Ch. 6.4 - Use the arc length formula from Exercise 26 to...Ch. 6.4 - Use the arc length formula from Exercise 26 to...Ch. 6.4 - Use the arc length formula from Exercise 26 to...Ch. 6.4 - Use the arc length formula from Exercise 26 to...Ch. 6.4 - (a) Show that the total arc length of the ellipse...Ch. 6.4 - Show that the total arc length of the ellipse...Ch. 6.4 - Writing In our discussion of Arc Length Problem...Ch. 6.5 - If f is a smooth, nonnegative function on a,b ,...Ch. 6.5 - The lateral area of the frustum with slant height...Ch. 6.5 - Anintegral expression for the area of the surface...Ch. 6.5 - An integral expression for the area of the surface...Ch. 6.5 - Find the area of the surface generated by...Ch. 6.5 - Prob. 2ESCh. 6.5 - Find the area of the surface generated by...Ch. 6.5 - Find the area of the surface generated by...Ch. 6.5 - Find the area of the surface generated by...Ch. 6.5 - Find the area of the surface generated by...Ch. 6.5 - Find the area of the surface generated by...Ch. 6.5 - Find the area of the surface generated by...Ch. 6.5 - Use a CAS to find the exact area of the surface...Ch. 6.5 - Use a CAS to find the exact area of the surface...Ch. 6.5 - Use a CAS to find the exact area of the surface...Ch. 6.5 - Prob. 12ESCh. 6.5 - Use a CAS or a calculating utility with a...Ch. 6.5 - Prob. 14ESCh. 6.5 - Use a CAS or a calculating utility with a...Ch. 6.5 - Prob. 16ESCh. 6.5 - True-False Determine whether the statement is true...Ch. 6.5 - True-False Determine whether the statement is true...Ch. 6.5 - True-False Determine whether the statement is true...Ch. 6.5 - True-False Determine whether the statement is true...Ch. 6.5 - Approximate the area of the surface using Formula...Ch. 6.5 - Approximate the area of the surface using Formula...Ch. 6.5 - Assume that y=fx is a smooth curve on the interval...Ch. 6.5 - Would it be circular reasoning to use Definition...Ch. 6.5 - Show that the area of the surface of a sphere of...Ch. 6.5 - The accompanying figure shows a spherical cap of...Ch. 6.5 - The portion of a sphere that is cut by two...Ch. 6.5 - Let y=fx be a smooth curve on the interval a,b and...Ch. 6.5 - Prob. 29ESCh. 6.5 - Let y=fx be a smooth curve on a,b and assume that...Ch. 6.6 - If a constant force of 5lb moves an object 10ft ,...Ch. 6.6 - A newton-meter is also called a . A...Ch. 6.6 - Suppose that an object moves in the positive...Ch. 6.6 - A force Fx=102xN applied in the positive x...Ch. 6.6 - Prob. 1ESCh. 6.6 - Prob. 2ESCh. 6.6 - A variable force Fx in Exercise 2 , consider the...Ch. 6.6 - Suppose that a variable force Fx is applied in the...Ch. 6.6 - A constant force of 10lb in the positive x-...Ch. 6.6 - A spring exerts a force of 6N when it is stretched...Ch. 6.6 - A spring exerts a force of 100N when it is...Ch. 6.6 - A spring whose natural length is 15cm exerts a...Ch. 6.6 - Assume that 10ftlb of work is required to stretch...Ch. 6.6 - True-False Determine whether the statement is true...Ch. 6.6 - True-False Determine whether the statement is true...Ch. 6.6 - True-False Determine whether the statement is true...Ch. 6.6 - True-False Determine whether the statement is true...Ch. 6.6 - A cylindrical tank of radius 5ft and height 9ft is...Ch. 6.6 - Solve Exercise 14 assuming that the tank is...Ch. 6.6 - A cone-shaped water reservoir is 20ft in diameter...Ch. 6.6 - The vat shown in the accompanying figure contains...Ch. 6.6 - The cylindrical tank shown in the accompanying...Ch. 6.6 - A swimming pool is built in the shape of a...Ch. 6.6 - How much work is required to fill the swimming...Ch. 6.6 - A 100ft length of steel chain weighing 15lb/ft is...Ch. 6.6 - A 3lb bucket containing 20lb of water is hanging...Ch. 6.6 - A rocket weighing 3 tons is filled with 40 tons of...Ch. 6.6 - It follows from Coulomb’s law in physics that...Ch. 6.6 - It follows from Newton’s Law of Universal...Ch. 6.6 - (a) The formula x=k/x2 in Exercise 25 is...Ch. 6.6 - The world’s first commercial high-speed magnetic...Ch. 6.6 - Assume that a Mars probe of mass m=2.00103kg is...Ch. 6.6 - On August 10,1972 a meteorite with an estimated...Ch. 6.7 - Prob. 1QCECh. 6.7 - A homogeneous lamina of mass M and density ...Ch. 6.7 - Let R be the region between the graphs of y=x2 and...Ch. 6.7 - If the region R in Quick Check Exercise 3 is used...Ch. 6.7 - Masses m1=5,m2=10, and m3=20 are positioned on a...Ch. 6.7 - Masses m1=10,m2=3,m3=4, and m are positioned on a...Ch. 6.7 - Find the centroid of the region by inspection and...Ch. 6.7 - Find the centroid of the region by inspection and...Ch. 6.7 - Find the centroid of the region by inspection and...Ch. 6.7 - Find the centroid of the region by inspection and...Ch. 6.7 - Find the centroid of the region.Ch. 6.7 - Find the centroid of the region.Ch. 6.7 - Find the centroid of the region.Ch. 6.7 - Find the centroid of the region.Ch. 6.7 - Find the centroid of the region. The triangle with...Ch. 6.7 - Find the centroid of the region. The region...Ch. 6.7 - Find the centroid of the region. The region...Ch. 6.7 - Find the centroid of the region. The region...Ch. 6.7 - Find the centroid of the region. The region...Ch. 6.7 - Find the centroid of the region. The region...Ch. 6.7 - Use symmetry considerations to argue that the...Ch. 6.7 - Use symmetry considerations to argue that the...Ch. 6.7 - Find the mass and center of gravity of the lamina...Ch. 6.7 - Find the mass and center of gravity of the lamina...Ch. 6.7 - Find the mass and center of gravity of the lamina...Ch. 6.7 - Use a CAS to find the mass and center of gravity...Ch. 6.7 - Use a CAS to find the mass and center of gravity...Ch. 6.7 - Use a CAS to find the mass and center of gravity...Ch. 6.7 - Determine whether the statement is true or false....Ch. 6.7 - Determine whether the statement is true or false....Ch. 6.7 - Determine whether the statement is true or false....Ch. 6.7 - Determine whether the statement is true or false....Ch. 6.7 - Find the centroid of the triangle with vertices...Ch. 6.7 - Prove that the centroid of a triangle is the point...Ch. 6.7 - Find the centroid of the isosceles trapezoid with...Ch. 6.7 - Prove that the centroid of a parallelogram is the...Ch. 6.7 - Use the Theorem of Pappus and the fact that the...Ch. 6.7 - Use the Theorem of Pappus and the fact that the...Ch. 6.7 - Use the Theorem of Pappus to find the volume of...Ch. 6.7 - Use the Theorem of Pappus to find the centroid of...Ch. 6.8 - The pressure unit equivalent to a newton per...Ch. 6.8 - Given that the weight density of water is 9810N/m3...Ch. 6.8 - Suppose that a flat surface is immersed vertically...Ch. 6.8 - A rectangular plate 2m wide and 3m high is...Ch. 6.8 - In this exercise set, refer to Table 6.8.2 for...Ch. 6.8 - (a) Find the force (in N ) on the deck of a sunken...Ch. 6.8 - The flat surfaces shown are submerged vertically...Ch. 6.8 - The flat surfaces shown are submerged vertically...Ch. 6.8 - The flat surfaces shown are submerged vertically...Ch. 6.8 - The flat surfaces shown are submerged vertically...Ch. 6.8 - The flat surfaces shown are submerged vertically...Ch. 6.8 - The flat surfaces shown are submerged vertically...Ch. 6.8 - Suppose that a flat surface is immersed vertically...Ch. 6.8 - An oil tank is shaped like a right circular...Ch. 6.8 - A square plate of side a feet is dipped in a...Ch. 6.8 - True-False Determine whether the statement is true...Ch. 6.8 - True-False Determine whether the statement is true...Ch. 6.8 - True-False Determine whether the statement is true...Ch. 6.8 - True-False Determine whether the statement is true...Ch. 6.8 - Formula 8 gives the fluid force on a flat surface...Ch. 6.8 - Prob. 17ESCh. 6.8 - Formula 8 gives the fluid force on a flat surface...Ch. 6.8 - Prob. 19ESCh. 6.8 - An observation window on a submarine is a square...Ch. 6.8 - (a) Show: If the submarine in Exercise 20 descends...Ch. 6.8 - (a) Let D=Da denote a disk of radius a submerged...Ch. 6.9 - coshx=sinhx=tanhx=Ch. 6.9 - Complete the table.Ch. 6.9 - Prob. 3QCECh. 6.9 - ddxcoshx=ddxsinhx=ddxtanhx=Ch. 6.9 - coshxdx=sinhxdx=tanhxdx=Ch. 6.9 - ddxcosh1x=ddxsinh1x=ddxtanh1x=Ch. 6.9 - Approximate the expression to four decimal places....Ch. 6.9 - Find the exact numerical value of each expression,...Ch. 6.9 - In each part, rewrite the expression as a ratio of...Ch. 6.9 - In each part, a value for one of the hyperbolic...Ch. 6.9 - Obtain the derivative formulas for cschx,sechx,...Ch. 6.9 - Find the derivatives of cosh1x and tanh1x , and...Ch. 6.9 - Find the derivatives of sinh1x,cosh1x,tanh1x by...Ch. 6.9 - Find dy/dx . y=sinh4x8Ch. 6.9 - Find dy/dx . y=coshx4Ch. 6.9 - Find dy/dx . y=cothlnxCh. 6.9 - Find dy/dx 12. y=ln(tanh5x)Ch. 6.9 - Find dy/dx . y=csch1/xCh. 6.9 - Find dy/dx . y=seche2xCh. 6.9 - Find dy/dx . y=4x+cosh25xCh. 6.9 - Find dy/dx 16. y=sinh4(3x)Ch. 6.9 - Find dy/dx . y=x3tanh2xCh. 6.9 - Find dy/dx . y=sinhcos3xCh. 6.9 - Find dy/dx . y=sinh113xCh. 6.9 - Find dy/dx . y=sinh11/xCh. 6.9 - Prob. 21ESCh. 6.9 - Find dy/dx . y=cosh1sinh1xCh. 6.9 - Find dy/dx . y=coth1x2Ch. 6.9 - Prob. 25ESCh. 6.9 - Prob. 26ESCh. 6.9 - Prob. 27ESCh. 6.9 - Prob. 28ESCh. 6.9 - Evaluate the integrals. sinh6xcoshxdxCh. 6.9 - Evaluate the integrals. 30. cosh(3x+2)dxCh. 6.9 - Evaluate the integrals. tanhxsech2xdxCh. 6.9 - Evaluate the integrals. csch23xdxCh. 6.9 - Evaluate the integrals. tanhxdxCh. 6.9 - Prob. 34ESCh. 6.9 - Evaluate the integrals. ln2ln3tanhxsech3xdxCh. 6.9 - Evaluate the integrals. 0ln3exexex+exdxCh. 6.9 - Evaluate the integrals. dx1+9x2Ch. 6.9 - Evaluate the integrals. dxx22x2Ch. 6.9 - Evaluate the integrals. dx1e2xx0Ch. 6.9 - Evaluate the integrals. sind1+cos2Ch. 6.9 - Evaluate the integrals. dxx1+4x2Ch. 6.9 - Prob. 42ESCh. 6.9 - Prob. 43ESCh. 6.9 - Evaluate the integrals. 03dtt2+1Ch. 6.9 - True-False Determine whether the statement is true...Ch. 6.9 - True-False Determine whether the statement is true...Ch. 6.9 - True-False Determine whether the statement is true...Ch. 6.9 - True-False Determine whether the statement is true...Ch. 6.9 - Find the area enclosed by y=sinh2x,y=0, and x=ln3...Ch. 6.9 - Find the volume of the solid that is generated...Ch. 6.9 - Find the arc length of the catenary y=coshx...Ch. 6.9 - Find the arc length of the catenary y=acoshx/a...Ch. 6.9 - In parts (a)-(f) find the limits, and confirm that...Ch. 6.9 - Explain how to obtain the asymptotes for y=tanhx...Ch. 6.9 - Prove that sinhx is an odd function of x and that...Ch. 6.9 - Prove the identities. (a) coshx+sinhx=ex (b)...Ch. 6.9 - Prove the identities. (a) 1tanh2x=sech2x (b)...Ch. 6.9 - Prove: (a) cosh1x=lnx+x21,x1 (b)...Ch. 6.9 - Use Exercise 60 to obtain the derivative formulas...Ch. 6.9 - Prove:...Ch. 6.9 - Use Exercise 62 to express the integral dy1u2...Ch. 6.9 - Show that (a) ddxsech1x=1x1+x2 (b)...Ch. 6.9 - In each part, find the limit. (a) limx+cosh1xlnx...Ch. 6.9 - Use the first and second derivatives to show that...Ch. 6.9 - The integration formulas for 1/u2a2 in Theorem...Ch. 6.9 - Show that sinhx+coshxn=sinhnx+coshnx .Ch. 6.9 - Show that aaetxdx=2sinattCh. 6.9 - A cable is suspended between two poles as shown in...Ch. 6.9 - These exercises refer to the hanging cable...Ch. 6.9 - These exercises refer to the hanging cable...Ch. 6.9 - The design of the Gateway Arch in St. Louis,...Ch. 6.9 - Suppose that a hollow tube rotates with a constant...Ch. 6.9 - The accompanying figure shows a person pulling a...Ch. 6.9 - The death rate for victims of a plague during the...Ch. 6 - Describe the method of slicing for finding...Ch. 6 - State an integral formula for finding a volume by...Ch. 6 - State an integral formula for finding the arc...Ch. 6 - State an integral formula for the work W done by a...Ch. 6 - State an integral formula for the fluid force F...Ch. 6 - Let R be the region in the first quadrant enclosed...Ch. 6 - (a) Set up a sum of definite integrals that...Ch. 6 - The accompanying figure shows velocity versus time...Ch. 6 - Let R be the region enclosed by the curves...Ch. 6 - A football has the shape of the solid generated by...Ch. 6 - Find the volume of the solid whose base is the...Ch. 6 - Find the arc length in the second quadrant of the...Ch. 6 - Let C be the curve y=ex between x=0 and x=ln10 ....Ch. 6 - Find the area of the surface generated by...Ch. 6 - Consider the solid generated by revolving the...Ch. 6 - (a) A spring exerts a force of 0.5N when stretched...Ch. 6 - A boat is anchored so that the anchor is 150ft...Ch. 6 - Find the centroid of the region. The region...Ch. 6 - Find the centroid of the region. The upper half of...Ch. 6 - In each part, set up, but do not evaluate, an...Ch. 6 - Show that for any constant a, the function...Ch. 6 - In each part, prove the identity. (a)...
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- 1. Evaluate the integral ↓ f(x, y)dxdy, of function f R² →R over the domain DC R2, where: f(x, y) = 2x + y and D is the is the triangle with vertices (0, -1), (1,0) and (0,2). Hint. Represent D in the form D = {(x, y) = R² : x = (a, b), g(x) < y < h(x)} for some aarrow_forward2. (i) Describe with a sketch the trace of the following curve in R² (t) = (t² sin(t), t2 cos(t)), (€ -2,2]). [15 Marks] (ii) Find the length of this curve. [25 Marks]arrow_forwardCalculus Problem: Please help . thank you. Find f'(x)arrow_forward5. mit answer urces Use Simpson's Rule and all the data in the following table to estimate the value of the 31 integral f(x) dx. 25 25 26 27 28 29 30 31 f(x) 4 44 4 -9 -2 9 2 5 (Round your answer to within two decimal places if necessary, but do not round until your final computation.) Simpson's Rule Approximation: PROGRES Score Completi 30 i Submit answer T The Weather Channel UP DELL FB F4 F5 F9 9. F10arrow_forwardHelparrow_forwardneed helparrow_forwardFind the value of the integral: ∬xysin(x)cos(x) dA where the region R lies between the graphs of sine and cosine functions and between 45 and 225 degrees along the xxx-axis. a) Determine the correct limits and write the order of integration that you consider appropriate.b) Solve the integral, providing a detailed step-by-step solution.arrow_forwardFind the value of the following indefinite integral: (see image)arrow_forwardFind the volume of the following figure using double integrals in polar coordinates. The base of the figure is at z=0, and the figure is the lower part of the cone defined by the equation: a2z2- b2x2- b2y2-2 a2bz+ a2b2arrow_forwardShow all steps. Correct answer is 37.6991118arrow_forwardFind the flux F(x, y, z) = xi + 2yj +4zk, S is the cube with vertices (1, 1, 1), (-1, -1, -1)arrow_forwardB-Solve the D.E of the following: 1- y+3y+2fy dt = f(t) for y(0)-1 if f(t) is the function whose graph is shown below 2- y" +4y = u(t) for y(0)-y'(0)-0 3- y"+4y+13y=e-2t sin3t 1 2 for y(0)-1 and y'(0)=-2arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
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