Calculus, Early Transcendentals
9th Edition
ISBN: 9781337613927
Author: Stewart
Publisher: CENGAGE L
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Chapter 10.2, Problem 1E
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(3) (20 points) Let F(x, y, z) = (y, z, x²z). Define
E = {(x, y, z) | x² + y² ≤ z ≤ 1, x ≤ 0}.
(a) (2 points) Calculate the divergence V. F.
(b) (4 points) Let D = {(x, y) | x² + y² ≤ 1, x ≤ 0} Without calculation, show that
the triple integral
√ (V · F) dV = √ 2²(1.
= x²(1 − x² - y²) dA.
E
(2) (22 points) Let F(x, y, z) = (x sin y, cos y, ―xy).
(a) (2 points) Calculate V. F.
(b) (6 points) Given a vector field
is everywhere defined with V
G₁(x, y, z) = *
G2(x, y, z) = −
G3(x, y, z) = 0.
0
0
F(x, y, z) = (F₁(x, y, z), F₂(x, y, z), F(x, y, z)) that
F = 0, let G = (G1, G2, G3) where
F₂(x,
y,
y, t) dt
- √ F³(x, t, 0) dt,
*
F1(x,
y, t) dt,
t) dt - √ F
Calculate G for the vector field F(x, y, z) = (x sin y, cos y, -xy).
Evaluate the following integral over the Region R.
(Answer accurate to 2 decimal places).
√ √(x + y) A
R
R = {(x, y) | 25 < x² + y² ≤ 36, x < 0}
Hint: The integral and Region is defined in rectangular coordinates.
Chapter 10 Solutions
Calculus, Early Transcendentals
Ch. 10.1 - Prob. 1ECh. 10.1 - For the given parametric equations, find the...Ch. 10.1 - Sketch the curve by using the parametric equations...Ch. 10.1 - Sketch the curve by using the parametric equations...Ch. 10.1 - Prob. 7ECh. 10.1 - Prob. 8ECh. 10.1 - Prob. 9ECh. 10.1 - Prob. 10ECh. 10.1 - Prob. 11ECh. 10.1 - Prob. 12E
Ch. 10.1 - Prob. 16ECh. 10.1 - Prob. 17ECh. 10.1 - Prob. 22ECh. 10.1 - (a) Eliminate the parameter to find a Cartesian...Ch. 10.1 - Describe the motion of a particle with position...Ch. 10.1 - Describe the motion of a particle with position...Ch. 10.1 - Describe the motion of a particle with position...Ch. 10.1 - Describe the motion of a particle with position...Ch. 10.1 - Suppose a curve is given by the parametric...Ch. 10.1 - Match the graphs of the parametric equations x =...Ch. 10.1 - Prob. 31ECh. 10.1 - Prob. 32ECh. 10.1 - Prob. 33ECh. 10.1 - Graph the curve x = y 2 sin y.Ch. 10.1 - Graph the curves y = x3 4x and x = y3 4y and...Ch. 10.1 - (a) Show that the parametric equations x = x1 +...Ch. 10.1 - Use a graphing device and the result of Exercise...Ch. 10.1 - Find parametric equations for the position of a...Ch. 10.1 - Prob. 40ECh. 10.1 - Find parametric equations for the path of a...Ch. 10.1 - (a) Find parametric equations for the ellipse...Ch. 10.1 - Use a graphing calculator or computer to reproduce...Ch. 10.1 - Use a graphing calculator or computer to reproduce...Ch. 10.1 - Derive Equations 1 for the case /2Ch. 10.1 - Let P be a point at a distance d from the center...Ch. 10.1 - If a and b are fixed numbers, find parametric...Ch. 10.1 - If a and b are fixed numbers, find parametric...Ch. 10.1 - A curve, called a witch of Maria Agnesi, consists...Ch. 10.1 - (a) Find parametric equations for the set of all...Ch. 10.1 - Investigate the family of curves defined by the...Ch. 10.1 - The swallowtail catastrophe curves are defined by...Ch. 10.1 - Graph several members of the family of curves with...Ch. 10.1 - Graph several members of the family of curves x =...Ch. 10.1 - Prob. 63ECh. 10.1 - Prob. 64ECh. 10.2 - Find dx/dt,dy/dt, and dy/dx. 1. x=2t3+3t,y=4t5t2Ch. 10.2 - Find dx/dt,dy/dt, and dy/dx. 2. x=tlnt,y=t2t2Ch. 10.2 - Prob. 3ECh. 10.2 - Prob. 4ECh. 10.2 - Find the slope of the tangent to the parametric...Ch. 10.2 - Find the slope of the tangent to the parametric...Ch. 10.2 - Find an equation of the tangent to the curve at...Ch. 10.2 - Find an equation of the tangent to the curve at...Ch. 10.2 - Find an equation of the tangent to the curve at...Ch. 10.2 - Find an equation of the tangent to the curve at...Ch. 10.2 - Find an equation of the tangent to the curve at...Ch. 10.2 - Find an equation of the tangent to the curve at...Ch. 10.2 - Find an equation of the tangent to the curve at...Ch. 10.2 - Find an equation of the tangent to the curve at...Ch. 10.2 - Find dy/dx and d2y/dx2. For which values of t is...Ch. 10.2 - Find dy/dx and d2y/dx2. For which values of t is...Ch. 10.2 - Find dy/dx and d2y/dx2. For which values of t is...Ch. 10.2 - Find dy/dx and d2y/dx2. For which values of t is...Ch. 10.2 - Find dy/dx and d2y/dx2. For which values of t is...Ch. 10.2 - Find dy/dx and d2y/dx2. For which values of t is...Ch. 10.2 - Find the points on the curve where the tangent is...Ch. 10.2 - Find the points on the curve where the tangent is...Ch. 10.2 - Find the points on the curve where the tangent is...Ch. 10.2 - Find the points on the curve where the tangent is...Ch. 10.2 - Use a graph to estimate the coordinates of the...Ch. 10.2 - Prob. 26ECh. 10.2 - Graph the curve in a viewing rectangle that...Ch. 10.2 - Graph the curve in a viewing rectangle that...Ch. 10.2 - Show that the curve x = cos t, y = sin t cos t has...Ch. 10.2 - Prob. 30ECh. 10.2 - At what point(s) on the curve x = 3t2 + 1, y = t3 ...Ch. 10.2 - Find equations of the tangents to the curve x =...Ch. 10.2 - Use the parametric equations of an ellipse, x = a...Ch. 10.2 - Let be the region enclosed by the loop of the...Ch. 10.2 - Find the exact length of the curve. 42. x = et t,...Ch. 10.2 - Find the exact length of the curve. 43. x = t sin...Ch. 10.2 - Find the exact length of the curve. 44. x = 3 cos...Ch. 10.2 - Graph the curve and find its exact length. 45. x =...Ch. 10.2 - Graph the curve and find its exact length. 46....Ch. 10.2 - Graph the curve x = sin t + sin 1.5t, y = cos t...Ch. 10.2 - Find the length of the loop of the curve x = 3t ...Ch. 10.2 - Prob. 57ECh. 10.2 - Prob. 58ECh. 10.2 - Prob. 59ECh. 10.2 - Prob. 60ECh. 10.2 - The curve shown in the figure is the astroid...Ch. 10.2 - Set up an integral that represents the area of the...Ch. 10.2 - Prob. 68ECh. 10.2 - Prob. 69ECh. 10.2 - Prob. 70ECh. 10.2 - Find the exact area of the surface obtained by...Ch. 10.2 - Find the exact area of the surface obtained by...Ch. 10.2 - Find the exact area of the surface obtained by...Ch. 10.2 - Prob. 74ECh. 10.2 - Find the surface area generated by rotating the...Ch. 10.2 - Prob. 76ECh. 10.2 - Prob. 77ECh. 10.2 - Prob. 78ECh. 10.2 - A cow is tied to a silo with radius r by a rope...Ch. 10.3 - Plot the point whose polar coordinates are given....Ch. 10.3 - Prob. 2ECh. 10.3 - Plot the point whose polar coordinates are given....Ch. 10.3 - Plot the point whose polar coordinates are given....Ch. 10.3 - Prob. 5ECh. 10.3 - Prob. 6ECh. 10.3 - Prob. 7ECh. 10.3 - Prob. 8ECh. 10.3 - Sketch the region in the plane consisting of...Ch. 10.3 - Prob. 10ECh. 10.3 - Sketch the region in the plane consisting of...Ch. 10.3 - Prob. 12ECh. 10.3 - Find the distance between the points with polar...Ch. 10.3 - Prob. 14ECh. 10.3 - Identify the curve by finding a Cartesian equation...Ch. 10.3 - Prob. 16ECh. 10.3 - Identify the curve by finding a Cartesian equation...Ch. 10.3 - Identify the curve by finding a Cartesian equation...Ch. 10.3 - Identify the curve by finding a Cartesian equation...Ch. 10.3 - Prob. 20ECh. 10.3 - Prob. 21ECh. 10.3 - Prob. 22ECh. 10.3 - Prob. 23ECh. 10.3 - Prob. 24ECh. 10.3 - Prob. 25ECh. 10.3 - Find a polar equation for the curve represented by...Ch. 10.3 - Prob. 27ECh. 10.3 - Prob. 28ECh. 10.3 - Prob. 29ECh. 10.3 - Sketch the curve with the given polar equation by...Ch. 10.3 - Sketch the curve with the given polar equation by...Ch. 10.3 - Sketch the curve with the given polar equation by...Ch. 10.3 - Prob. 36ECh. 10.3 - Sketch the curve with the given polar equation by...Ch. 10.3 - Prob. 38ECh. 10.3 - Sketch the curve with the given polar equation by...Ch. 10.3 - Sketch the curve with the given polar equation by...Ch. 10.3 - Sketch the curve with the given polar equation by...Ch. 10.3 - Sketch the curve with the given polar equation by...Ch. 10.3 - Sketch the curve with the given polar equation by...Ch. 10.3 - Sketch the curve with the given polar equation by...Ch. 10.3 - Sketch the curve with the given polar equation by...Ch. 10.3 - Sketch the curve with the given polar equation by...Ch. 10.3 - Sketch the curve with the given polar equation by...Ch. 10.3 - Prob. 48ECh. 10.3 - Sketch the curve with the given polar equation by...Ch. 10.3 - Prob. 50ECh. 10.3 - Prob. 51ECh. 10.3 - Show that the curve r = 2 csc (also a conchoid)...Ch. 10.3 - Show that the curve r = sin tan (called a...Ch. 10.3 - Sketch the curve (x2 + y2)3 = 4x2y2.Ch. 10.3 - Prob. 57ECh. 10.3 - Show that the curves r = a sin and r = a cos ...Ch. 10.3 - Prob. 59ECh. 10.3 - Prob. 60ECh. 10.3 - Prob. 61ECh. 10.3 - Prob. 62ECh. 10.3 - Prob. 63ECh. 10.3 - Graph the polar curve. Choose a parameter interval...Ch. 10.3 - Prob. 65ECh. 10.3 - Prob. 66ECh. 10.3 - Prob. 67ECh. 10.3 - Prob. 68ECh. 10.4 - Find the area of the region that is bounded by the...Ch. 10.4 - Find the area of the region that is bounded by the...Ch. 10.4 - Find the area of the region that is bounded by the...Ch. 10.4 - Find the area of the region that is bounded by the...Ch. 10.4 - Find the area of the shaded region. 5.Ch. 10.4 - Find the area of the shaded region. 6.Ch. 10.4 - Find the area of the shaded region. 7.Ch. 10.4 - Find the area of the shaded region. 8.Ch. 10.4 - Sketch the curve and find the area that it...Ch. 10.4 - Sketch the curve and find the area that it...Ch. 10.4 - Prob. 11ECh. 10.4 - Prob. 12ECh. 10.4 - Graph the curve and find the area that it...Ch. 10.4 - Prob. 14ECh. 10.4 - Prob. 15ECh. 10.4 - Prob. 16ECh. 10.4 - Find the area of the region enclosed by one loop...Ch. 10.4 - Find the area of the region enclosed by one loop...Ch. 10.4 - Find the area of the region enclosed by one loop...Ch. 10.4 - Find the area of the region enclosed by one loop...Ch. 10.4 - Find the area of the region enclosed by one loop...Ch. 10.4 - Find the area enclosed by the loop of the...Ch. 10.4 - Find the area of the region that lies inside the...Ch. 10.4 - Find the area of the region that lies inside the...Ch. 10.4 - Find the area of the region that lies inside the...Ch. 10.4 - Find the area of the region that lies inside the...Ch. 10.4 - Find the area of the region that lies inside the...Ch. 10.4 - Find the area of the region that lies inside the...Ch. 10.4 - Find the area of the region that lies inside both...Ch. 10.4 - Find the area of the region that lies inside both...Ch. 10.4 - Find the area of the region that lies inside both...Ch. 10.4 - Find the area of the region that lies inside both...Ch. 10.4 - Find the area of the region that lies inside both...Ch. 10.4 - Prob. 34ECh. 10.4 - Find the area inside the larger loop and outside...Ch. 10.4 - Find the area between a large loop and the...Ch. 10.4 - Find all points of intersection of the given...Ch. 10.4 - Find all points of intersection of the given...Ch. 10.4 - Find all points of intersection of the given...Ch. 10.4 - Prob. 40ECh. 10.4 - Find all points of intersection of the given...Ch. 10.4 - Prob. 42ECh. 10.4 - Prob. 43ECh. 10.4 - The points of intersection of the cardioid r = 1 +...Ch. 10.4 - When recording live performances, sound engineers...Ch. 10.4 - Find the exact length of the polar curve. 45. r =...Ch. 10.4 - Find the exact length of the polar curve. 47. r =...Ch. 10.4 - Find the exact length of the polar curve. 48. r =...Ch. 10.4 - Find the exact length of the curve. Use a graph to...Ch. 10.4 - Find the exact length of the curve. Use a graph to...Ch. 10.4 - Use a calculator to find the length of the curve...Ch. 10.4 - Prob. 60ECh. 10.4 - Prob. 61ECh. 10.4 - Prob. 62ECh. 10.4 - Prob. 65ECh. 10.4 - Prob. 67ECh. 10.4 - Find the points on the given curve where the...Ch. 10.5 - Find the vertex, focus, and directrix of the...Ch. 10.5 - Prob. 3ECh. 10.5 - Prob. 6ECh. 10.5 - Find the vertex, focus, and directrix of the...Ch. 10.5 - Find the vertex, focus, and directrix of the...Ch. 10.5 - Find an equation of the parabola. Then find the...Ch. 10.5 - Find an equation of the parabola. Then find the...Ch. 10.5 - Prob. 11ECh. 10.5 - Prob. 17ECh. 10.5 - Prob. 18ECh. 10.5 - Prob. 19ECh. 10.5 - Prob. 20ECh. 10.5 - Prob. 21ECh. 10.5 - Prob. 22ECh. 10.5 - Prob. 23ECh. 10.5 - Prob. 24ECh. 10.5 - Prob. 25ECh. 10.5 - Prob. 26ECh. 10.5 - Identify the type of conic section whose equation...Ch. 10.5 - Identify the type of conic section whose equation...Ch. 10.5 - Prob. 29ECh. 10.5 - Prob. 30ECh. 10.5 - Prob. 31ECh. 10.5 - Identify the type of conic section whose equation...Ch. 10.5 - Prob. 33ECh. 10.5 - Find an equation for the conic that satisfies the...Ch. 10.5 - Prob. 35ECh. 10.5 - Prob. 36ECh. 10.5 - Prob. 37ECh. 10.5 - Prob. 38ECh. 10.5 - Prob. 39ECh. 10.5 - Find an equation for the conic that satisfies the...Ch. 10.5 - Prob. 41ECh. 10.5 - Prob. 42ECh. 10.5 - Prob. 43ECh. 10.5 - Prob. 44ECh. 10.5 - Prob. 45ECh. 10.5 - Prob. 46ECh. 10.5 - Prob. 47ECh. 10.5 - Prob. 48ECh. 10.5 - Prob. 49ECh. 10.5 - Prob. 50ECh. 10.5 - The point in a lunar orbit nearest the surface of...Ch. 10.5 - A cross-section of a parabolic reflector is shown...Ch. 10.5 - The LORAN (LOng RAnge Navigation) radio navigation...Ch. 10.5 - Use the definition of a hyperbola to derive...Ch. 10.5 - Show that the function defined by the upper branch...Ch. 10.5 - Find an equation for the ellipse with foci (1, 1)...Ch. 10.5 - Determine the type of curve represented by the...Ch. 10.5 - Prob. 58ECh. 10.5 - Prob. 59ECh. 10.5 - Prob. 60ECh. 10.5 - Prob. 61ECh. 10.5 - Prob. 62ECh. 10.5 - Find the area of the region enclosed by the...Ch. 10.5 - Prob. 64ECh. 10.5 - Find the centroid of the region enclosed by the...Ch. 10.5 - Prob. 66ECh. 10.6 - Prob. 5ECh. 10.6 - Prob. 6ECh. 10.6 - Prob. 7ECh. 10.6 - Prob. 8ECh. 10.6 - Prob. 9ECh. 10.6 - Prob. 10ECh. 10.6 - Prob. 11ECh. 10.6 - Prob. 12ECh. 10.6 - Prob. 13ECh. 10.6 - Prob. 14ECh. 10.6 - Prob. 15ECh. 10.6 - Prob. 16ECh. 10.6 - Prob. 17ECh. 10.6 - Prob. 18ECh. 10.6 - Prob. 19ECh. 10.6 - Prob. 20ECh. 10.6 - Prob. 21ECh. 10.6 - Prob. 22ECh. 10.6 - Prob. 23ECh. 10.6 - Prob. 24ECh. 10.6 - Prob. 25ECh. 10.6 - Prob. 26ECh. 10.6 - Prob. 27ECh. 10.6 - Prob. 28ECh. 10.6 - Prob. 29ECh. 10.6 - Prob. 30ECh. 10.6 - Prob. 31ECh. 10.6 - Jupiter's orbit has eccentricity 0.048 and the...Ch. 10.6 - The orbit of Halleys comet, last seen in 1986 and...Ch. 10.6 - Prob. 34ECh. 10.6 - Prob. 35ECh. 10.6 - Prob. 36ECh. 10.6 - Prob. 37ECh. 10 - (a) What is a parametric curve? (b) How do you...Ch. 10 - Prob. 2CCCh. 10 - Prob. 3CCCh. 10 - Prob. 5CCCh. 10 - Prob. 6CCCh. 10 - Prob. 7CCCh. 10 - Prob. 8CCCh. 10 - Prob. 9CCCh. 10 - (a) What is the eccentricity of a conic section?...Ch. 10 - Determine whether the statement is true or false....Ch. 10 - Determine whether the statement is true or false....Ch. 10 - Determine whether the statement is true or false....Ch. 10 - Determine whether the statement is true or false....Ch. 10 - Prob. 6TFQCh. 10 - Determine whether the statement is true or false....Ch. 10 - Determine whether the statement is true or false....Ch. 10 - Determine whether the statement is true or false....Ch. 10 - Determine whether the statement is true or false....Ch. 10 - Determine whether the statement is true or false....Ch. 10 - Prob. 1ECh. 10 - Prob. 2ECh. 10 - Prob. 3ECh. 10 - Prob. 4ECh. 10 - Prob. 5ECh. 10 - Prob. 7ECh. 10 - Prob. 8ECh. 10 - Prob. 9ECh. 10 - Prob. 10ECh. 10 - Prob. 11ECh. 10 - Prob. 12ECh. 10 - Prob. 13ECh. 10 - Sketch the polar curve. 12. r = 3 + cos 3Ch. 10 - Prob. 15ECh. 10 - Prob. 16ECh. 10 - Prob. 17ECh. 10 - Prob. 18ECh. 10 - Prob. 19ECh. 10 - Prob. 20ECh. 10 - The curve with polar equation r = (sin )/ is...Ch. 10 - Prob. 23ECh. 10 - Prob. 24ECh. 10 - Prob. 25ECh. 10 - Prob. 26ECh. 10 - Prob. 27ECh. 10 - Prob. 28ECh. 10 - Prob. 29ECh. 10 - Prob. 30ECh. 10 - Prob. 31ECh. 10 - Prob. 32ECh. 10 - Find the area enclosed by the curve r2 = 9 cos 5.Ch. 10 - Prob. 34ECh. 10 - Prob. 35ECh. 10 - Prob. 36ECh. 10 - Find the area of the region that lies inside both...Ch. 10 - Find the area of the region that lies inside the...Ch. 10 - Prob. 39ECh. 10 - Prob. 40ECh. 10 - Prob. 41ECh. 10 - Prob. 42ECh. 10 - Prob. 45ECh. 10 - Prob. 46ECh. 10 - Prob. 47ECh. 10 - Prob. 48ECh. 10 - Prob. 49ECh. 10 - Prob. 50ECh. 10 - Prob. 51ECh. 10 - Prob. 52ECh. 10 - Prob. 53ECh. 10 - Prob. 54ECh. 10 - Prob. 55ECh. 10 - Prob. 56ECh. 10 - Prob. 57ECh. 10 - Prob. 58ECh. 10 - Prob. 59ECh. 10 - Prob. 61ECh. 10 - A curve called the folium of Descartes is defined...Ch. 10 - The outer circle in the figure has radius 1 and...Ch. 10 - Prob. 2PPCh. 10 - Prob. 3PPCh. 10 - Four bugs are placed at the four corners of a...Ch. 10 - Prob. 5PPCh. 10 - A circle C of radius 2r has its center at the...
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- Find the volume of the solid that lies under the paraboloid z = 81 - x² - y² and within the cylinder (x − 1)² + y² = 1. A plot of an example of a similar solid is shown below. (Answer accurate to 2 decimal places). Volume using Double Integral Paraboloid & Cylinder -3 Hint: The integral and region is defined in polar coordinates.arrow_forwardEvaluate the following integral over the Region R. (Answer accurate to 2 decimal places). √4(1–2² 4(1 - x² - y²) dA R 3 R = {(r,0) | 0 ≤ r≤ 2,0π ≤0≤¼˜}. Hint: The integral is defined in rectangular coordinates. The Region is defined in polar coordinates.arrow_forwardEvaluate the following integral over the Region R. (Answer accurate to 2 decimal places). R - 1 · {(r,0) | 1 ≤ r≤ 5,½π≤ 0<1π}. Hint: Be sure to convert to Polar coordinates. Use the correct differential for Polar Coordinates.arrow_forward
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