Essential Calculus: Early Transcendentals
2nd Edition
ISBN: 9781133112280
Author: James Stewart
Publisher: Cengage Learning
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Chapter 7.5, Problem 14E
To determine
To find: The area of the surface obtained when the curve is rotated about the y-axis.
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3.12 (B). A horizontal beam AB is 4 m
long and of constant flexural rigidity. It is
rigidly built-in at the left-hand end A and simply supported on a non-yielding support
at the right-hand end B. The beam carries Uniformly distributed vertical loading of
18 kN/m over its whole length, together with a vertical downward load of 10KN at
2.5 m from the end A. Sketch the S.F. and B.M. diagrams for the beam, indicating
all main values. Cl. Struct. E.] CS.F. 45,10,376 KN, B.M. 186, +36.15 kNm.7
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27 de febrero de 2025
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Chapter 7 Solutions
Essential Calculus: Early Transcendentals
Ch. 7.1 - Find the area of the shaded region.Ch. 7.1 - Prob. 2ECh. 7.1 - Find the area of the shaded region.Ch. 7.1 - Find the area of the shaded region.Ch. 7.1 - Prob. 5ECh. 7.1 - Sketch the region enclosed by the given curves....Ch. 7.1 - Prob. 7ECh. 7.1 - Sketch the region enclosed by the given curves....Ch. 7.1 - Prob. 9ECh. 7.1 - Prob. 10E
Ch. 7.1 - Sketch the region enclosed by the given curves and...Ch. 7.1 - 14. Sketch the region enclosed by the given curves...Ch. 7.1 - Sketch the region enclosed by the given curves and...Ch. 7.1 - Prob. 14ECh. 7.1 - Sketch the region enclosed by the given curves and...Ch. 7.1 - Sketch the region enclosed by the given curves and...Ch. 7.1 - Prob. 16ECh. 7.1 - Prob. 18ECh. 7.1 - Prob. 19ECh. 7.1 - Sketch the region enclosed by the given curves and...Ch. 7.1 - Sketch the region that lies between the curves y =...Ch. 7.1 - Graph the curves y = x2 x and y = x3 4x2 +3x on...Ch. 7.1 - Graph the region between the curves and use your...Ch. 7.1 - Graph the region between the curves and use your...Ch. 7.1 - Graph the region between the curves and use your...Ch. 7.1 - Graph the region between the curves and use your...Ch. 7.1 - Sketch the region in the xy-plane defined by the...Ch. 7.1 - Racing cars driven by Chris and Kelly are side by...Ch. 7.1 - The widths (in meters) of a kidney-shaped swimming...Ch. 7.1 - A cross-section of an airplane wing is shown....Ch. 7.1 - If the birth rate of a population is b(t) =...Ch. 7.1 - Two cars, A and B, start side by side and...Ch. 7.1 - Find a positive continuous function f such that...Ch. 7.1 - Find the area of the region bounded by the...Ch. 7.1 - Find the number b such that the line y = b divides...Ch. 7.1 - (a) Find the number a such that the line x = a...Ch. 7.1 - 59. Find the values of c such that the area of the...Ch. 7.1 - Suppose that 0 c /2. For what value of c is the...Ch. 7.1 - 61. For what values of m do the line y = mx and...Ch. 7.1 - 43. Find the area of the crescent-shaped region...Ch. 7.1 - A water storage tank has the shape of a cylinder...Ch. 7.2 - Find the volume of the solid obtained by rotating...Ch. 7.2 - Find the volume of the solid obtained by rotating...Ch. 7.2 - Find the volume of the solid obtained by rotating...Ch. 7.2 - Find the volume of the solid obtained by rotating...Ch. 7.2 - Find the volume of the solid obtained by rotating...Ch. 7.2 -
1–12 Find the volume of the solid obtained by...Ch. 7.2 - Find the volume of the solid obtained by rotating...Ch. 7.2 - Find the volume of the solid obtained by rotating...Ch. 7.2 - Find the volume of the solid obtained by rotating...Ch. 7.2 - Find the volume of the solid obtained by rotating...Ch. 7.2 - Find the volume of the solid obtained by rotating...Ch. 7.2 - Find the volume of the solid obtained by rotating...Ch. 7.2 - The region enclosed by the given curves is rotated...Ch. 7.2 - The region enclosed by the given curves is rotated...Ch. 7.2 - The region enclosed by the given curves is rotated...Ch. 7.2 - The region enclosed by the given curves is rotated...Ch. 7.2 - The region enclosed by the given curves is rotated...Ch. 7.2 - The region enclosed by the given curves is rotated...Ch. 7.2 - Set up an integral for the volume of the solid...Ch. 7.2 - Set up an integral for the volume of the solid...Ch. 7.2 - Set up an integral for the volume of the solid...Ch. 7.2 - Set up an integral for the volume of the solid...Ch. 7.2 - Use a graph to find approximate x-coordinates of...Ch. 7.2 - Use a graph to find approximate x-coordinates of...Ch. 7.2 - Each integral represents the volume of a solid....Ch. 7.2 - Each integral represents the volume of a solid....Ch. 7.2 - A CAT scan produces equally spaced cross-sectional...Ch. 7.2 - A log 10 m long is cut at 1-meter intervals and...Ch. 7.2 - 47. Find the volume of the described solid S.
A...Ch. 7.2 - Find the volume of the described solid S. A...Ch. 7.2 - Find the volume of the described solid S. A cap of...Ch. 7.2 - Find the volume of the described solid S. A...Ch. 7.2 - Find the volume of the described solid S. A...Ch. 7.2 - Find the volume of the described solid S. A...Ch. 7.2 - Find the volume of the described solid S. A...Ch. 7.2 - 54. Find the volume of the described solid S.
The...Ch. 7.2 - Find the volume of the described solid S. The base...Ch. 7.2 - Find the volume of the described solid S. The base...Ch. 7.2 - Find the volume of the described solid S. The base...Ch. 7.2 - Find the volume of the described solid S. The base...Ch. 7.2 - The base of S is the same base as in Exercise 42,...Ch. 7.2 - The base of S is a circular disk with radius r....Ch. 7.2 - (a) Set up an integral for the volume of a solid...Ch. 7.2 - A wedge is cut out of a circular cylinder of...Ch. 7.2 - (a) Cavalieris Principle states that if a family...Ch. 7.2 - Find the volume common to two circular cylinders,...Ch. 7.2 - Prob. 51ECh. 7.2 - A bowl is shaped like a hemisphere with diameter...Ch. 7.2 - A hole of radius r is bored through the middle of...Ch. 7.2 - A hole of radius r is bored through the center of...Ch. 7.2 - Some of the pioneers of calculus, such as Kepler...Ch. 7.3 - Let S be the solid obtained by rotating the region...Ch. 7.3 - Let S be the solid obtained by rotating the region...Ch. 7.3 - Use the method of cylindrical shells to find the...Ch. 7.3 - Use the method of cylindrical shells to find the...Ch. 7.3 - Use the method of cylindrical shells to find the...Ch. 7.3 - Use the method of cylindrical shells to find the...Ch. 7.3 - Use the method of cylindrical shells to find the...Ch. 7.3 - Let V be the volume of the solid obtained by...Ch. 7.3 - Use the method of cylindrical shells to find the...Ch. 7.3 - Use the method of cylindrical shells to find the...Ch. 7.3 - Use the method of cylindrical shells to find the...Ch. 7.3 - Use the method of cylindrical shells to find the...Ch. 7.3 - Use the method of cylindrical shells to find the...Ch. 7.3 - Prob. 14ECh. 7.3 - Use the method of cylindrical shells to find the...Ch. 7.3 - Use the method of cylindrical shells to find the...Ch. 7.3 - 17. Use the method of cylindrical shells to find...Ch. 7.3 - Use the method of cylindrical shells to find the...Ch. 7.3 - 15-20 Use the method of cylindrical shells to find...Ch. 7.3 - 15-20 Use the method of cylindrical shells to find...Ch. 7.3 - (a) Set up an integral for the volume of the solid...Ch. 7.3 - (a) Set up an integral for the volume of the solid...Ch. 7.3 - (a) Set up an integral for the volume of the solid...Ch. 7.3 - Prob. 24ECh. 7.3 - Prob. 25ECh. 7.3 - Prob. 26ECh. 7.3 - Use the Midpoint Rule with n = 4 to estimate the...Ch. 7.3 - Prob. 28ECh. 7.3 - Prob. 29ECh. 7.3 - Prob. 30ECh. 7.3 - Prob. 31ECh. 7.3 - Prob. 32ECh. 7.3 - Prob. 33ECh. 7.3 - Prob. 34ECh. 7.3 - Prob. 35ECh. 7.3 - Prob. 36ECh. 7.3 - Prob. 37ECh. 7.3 - Prob. 38ECh. 7.3 - Prob. 39ECh. 7.3 - 39-41 Use cylindrical shells to find the volume of...Ch. 7.3 - Use cylindrical shells to find the volume of the...Ch. 7.3 - Use cylindrical shells to find the volume of the...Ch. 7.3 - Prob. 43ECh. 7.4 - Use the arc length formula (3) to find the length...Ch. 7.4 - Use the arc length formula to find the length of...Ch. 7.4 - Set up an integral that represents the length of...Ch. 7.4 - Prob. 4ECh. 7.4 - Prob. 6ECh. 7.4 - Prob. 5ECh. 7.4 - Prob. 7ECh. 7.4 - Prob. 8ECh. 7.4 - Prob. 9ECh. 7.4 - Prob. 10ECh. 7.4 - Prob. 11ECh. 7.4 - Prob. 12ECh. 7.4 - Find the exact length of the curve. 15. y = ln(sec...Ch. 7.4 - Prob. 14ECh. 7.4 - Prob. 15ECh. 7.4 - Prob. 16ECh. 7.4 - Prob. 17ECh. 7.4 - Prob. 18ECh. 7.4 - Prob. 19ECh. 7.4 - Graph the curve and visually estimate its length....Ch. 7.4 - Prob. 21ECh. 7.4 - Prob. 22ECh. 7.4 - Prob. 23ECh. 7.4 - Use Simpsons Rule with n = 10 to estimate the arc...Ch. 7.4 - Prob. 25ECh. 7.4 - Prob. 27ECh. 7.4 - Prob. 28ECh. 7.4 - Prob. 29ECh. 7.4 - Prob. 30ECh. 7.4 - Find the arc length function for the curve...Ch. 7.4 - Prob. 32ECh. 7.4 - Prob. 33ECh. 7.4 - Prob. 35ECh. 7.4 - Prob. 34ECh. 7.4 - Prob. 36ECh. 7.5 - 41550-8.2-1E
1–6. (a) Set up an integral for the...Ch. 7.5 - Prob. 2ECh. 7.5 - Prob. 3ECh. 7.5 - Prob. 4ECh. 7.5 - Prob. 5ECh. 7.5 - Prob. 6ECh. 7.5 - 5-12 Find the exact area of the surface obtained...Ch. 7.5 - Prob. 8ECh. 7.5 - Prob. 9ECh. 7.5 - Prob. 10ECh. 7.5 - Prob. 11ECh. 7.5 - Find the exact area of the surface obtained by...Ch. 7.5 - Prob. 13ECh. 7.5 - Prob. 14ECh. 7.5 - Prob. 15ECh. 7.5 - The given curve is rotated about the y-axis. Find...Ch. 7.5 - Prob. 19ECh. 7.5 - Prob. 20ECh. 7.5 - Prob. 21ECh. 7.5 - Prob. 22ECh. 7.5 - Prob. 23ECh. 7.5 - Prob. 24ECh. 7.5 - Prob. 25ECh. 7.5 - Prob. 26ECh. 7.6 - Prob. 1ECh. 7.6 - Prob. 2ECh. 7.6 - Shown is the graph of a force function (in...Ch. 7.6 - Prob. 4ECh. 7.6 - Prob. 5ECh. 7.6 - A spring has a natural length of 20 cm. If a 25-N...Ch. 7.6 - Suppose that 2 J of work is needed to stretch a...Ch. 7.6 - If 6 J of work is needed to stretch a spring from...Ch. 7.6 - Prob. 9ECh. 7.6 - Show how to approximate the required work by a...Ch. 7.6 - Prob. 10ECh. 7.6 - Show how to approximate the required work by a...Ch. 7.6 - Show how to approximate the required work by a...Ch. 7.6 - Show how to approximate the required work by a...Ch. 7.6 - Show how to approximate the required work by a...Ch. 7.6 - 21. Show how to approximate the required work by a...Ch. 7.6 - The tank shown is full of water.
Find the work...Ch. 7.6 - (a) The tank shown is full of water. Given that...Ch. 7.6 - 29. When gas expands in a cylinder with radius r,...Ch. 7.6 - Prob. 20ECh. 7.6 - 33. (a) Newton’s Law of Gravitation states that...Ch. 7.6 - The Great Pyramid of King Khufu was built of...Ch. 7.6 - Use an improper integral and information from...Ch. 7.6 - Find the escape velocity v0 that is needed to...Ch. 7.6 - 2529 A vertical plate is submerged (or partially...Ch. 7.6 - 2529 A vertical plate is submerged (or partially...Ch. 7.6 - A vertical plate is submerged (or partially...Ch. 7.6 - 2529 A vertical plate is submerged (or partially...Ch. 7.6 - A vertical plate is submerged (or partially...Ch. 7.6 - A milk truck carries milk with density 64.6 lb/ft3...Ch. 7.6 - A trough is filled with a liquid of density 840...Ch. 7.6 - A vertical dam has a semicircular gate as shown in...Ch. 7.6 - Prob. 32ECh. 7.6 - A swimming pool is 20 ft wide and 40 ft long and...Ch. 7.6 - A metal plate was found submerged vertically in...Ch. 7.6 - Point-masses mi arc located on the x-axis as...Ch. 7.6 - The masses mi, are located at the points Pi. Find...Ch. 7.6 - 41550-8.3-24E
23–24 The masses mi, are located at...Ch. 7.6 - Sketch the region bounded by the curves, and...Ch. 7.6 - Sketch the region bounded by the curves, and...Ch. 7.6 - Sketch the region bounded by the curves, and...Ch. 7.6 - Sketch the region bounded by the curves, and...Ch. 7.6 - Find the centroid of the region bounded by the...Ch. 7.6 - Find the centroid of the region bounded by the...Ch. 7.6 - Find the centroid of the region bounded by the...Ch. 7.6 - Find the centroid of the region bounded by the...Ch. 7.6 - 4748 Calculate the moments Mx and My and the...Ch. 7.6 - 4748 Calculate the moments Mx and My and the...Ch. 7.6 - Prove that the centroid of any triangle is located...Ch. 7.6 - Find the centroid of the region shown, not by...Ch. 7.6 - Find the centroid of the region shown, not by...Ch. 7.6 - Use the Theorem of Pappus to find the volume of...Ch. 7.6 - Use the Theorem of Pappus to find the volume of...Ch. 7.6 - Use the Theorem of Pappus to find the volume of...Ch. 7.6 - 41550-8.3-50E
50. Let ℛ be the region that lies...Ch. 7.6 - Prove Formulas 13. 13....Ch. 7.7 - Psychologists interested in learning theory study...Ch. 7.7 - Prob. 21ECh. 7.7 - Match the differential equation with its direction...Ch. 7.7 - Prob. 23ECh. 7.7 - Match the differential equation with its direction...Ch. 7.7 - Prob. 25ECh. 7.7 - Prob. 26ECh. 7.7 - 41550-9.2-9E
9–10 Sketch a direction field for the...Ch. 7.7 - Sketch a direction field for the differential...Ch. 7.7 - Sketch the direction field of the differential...Ch. 7.7 - Sketch the direction field of the differential...Ch. 7.7 - Sketch the direction field of the differential...Ch. 7.7 - Sketch the direction field of the differential...Ch. 7.7 - Prob. 1ECh. 7.7 - Prob. 2ECh. 7.7 - Solve the differential equation. 3. xyy=x2+1Ch. 7.7 - Solve the differential equation. (y2 + xy2)y = 1Ch. 7.7 - Solve the differential equation. (y + sin y)y = x...Ch. 7.7 - Solve the differential equation. dyd=eysin2ysecCh. 7.7 - Prob. 7ECh. 7.7 - Prob. 8ECh. 7.7 - Find the solution of the differential equation...Ch. 7.7 - Prob. 10ECh. 7.7 - Prob. 11ECh. 7.7 - Find the solution of the differential equation...Ch. 7.7 - Prob. 13ECh. 7.7 - Prob. 14ECh. 7.7 - Prob. 15ECh. 7.7 - Prob. 16ECh. 7.7 - Prob. 17ECh. 7.7 - Prob. 18ECh. 7.7 - A sphere with radius 1 m has temperature 15C. It...Ch. 7.7 - A glucose solution is administered intravenously...Ch. 7.7 - A certain small country has 10 billion in paper...Ch. 7.7 - Write the solution of the logistic initial-value...Ch. 7.7 - A tank contains 1000 L of brine with 15 kg of...Ch. 7.7 - The air in a room with volume 180 m3 contains...Ch. 7.7 - A vat with 500 gallons of beer contains 4% alcohol...Ch. 7.7 - A tank contains 1000 L of pure water. Brine that...Ch. 7.7 - When a raindrop falls, it increases in size and so...Ch. 7.7 - 41550-9.3-50E
50. An object of mass m is moving...Ch. 7.7 - According to Newtons Law of Universal Gravitation,...Ch. 7.7 - The Pacific halibut fishery has been modeled by...Ch. 7.7 - One model for the spread of a rumor is that the...Ch. 7.7 - 41550-9.4-12E
12. Biologists stocked a lake with...Ch. 7.7 - (a) Show that if y satisfies the logistic equation...Ch. 7.7 - For a fixed value of M (say M = 10), the family of...Ch. 7 - Prob. 1RCCCh. 7 - Suppose that Sue runs faster than Kathy throughout...Ch. 7 - 3. (a) Suppose S is a solid with known...Ch. 7 - (a) What is the volume of a cylindrical shell? (b)...Ch. 7 - Suppose that you push a book across a 6-meter-long...Ch. 7 - 1. Find the area of the region bounded by the...Ch. 7 - Prob. 2RECh. 7 - Prob. 4RECh. 7 - Prob. 3RECh. 7 - Prob. 5RECh. 7 - Prob. 6RECh. 7 - Prob. 7RECh. 7 - Prob. 8RECh. 7 - Prob. 9RECh. 7 - Prob. 10RECh. 7 - Prob. 11RECh. 7 - Prob. 12RECh. 7 - Prob. 13RECh. 7 - Prob. 14RECh. 7 - Prob. 15RECh. 7 - Prob. 16RECh. 7 - Prob. 17RECh. 7 - Prob. 18RECh. 7 - Prob. 19RECh. 7 - Prob. 20RECh. 7 - Prob. 21RECh. 7 - Prob. 22RECh. 7 - Prob. 23RECh. 7 - Prob. 24RECh. 7 - A force of 30 N is required to maintain a spring...Ch. 7 - A 1600-lb elevator is suspended by a 200-ft cable...Ch. 7 - 29. A tank full of water has the shape of a...Ch. 7 - Let 1, be the region bounded by y = x2, y = 0, and...Ch. 7 - Prob. 5RCCCh. 7 - Prob. 6RCCCh. 7 - Prob. 8RCCCh. 7 - Prob. 9RCCCh. 7 - Prob. 10RCCCh. 7 - Prob. 25RECh. 7 - Prob. 26RECh. 7 - Prob. 27RECh. 7 - Prob. 28RECh. 7 - Use Simpson's Rule with n = 10 to estimate the...Ch. 7 - Use Simpsons Rule with n = 10 to estimate the area...Ch. 7 - Find the length of the curve y=1xt1dt1x16Ch. 7 - Prob. 32RECh. 7 - A gate in an irrigation canal is constructed in...Ch. 7 - A trough is filled with water and its vertical...Ch. 7 - Find the centroid of the region shown.Ch. 7 - Find the centroid of the region bounded by the...Ch. 7 - Find the centroid of the region bounded by the...Ch. 7 - Find the volume obtained when the circle of radius...Ch. 7 - 41550-8-18RE
18. Use the Theorem of Pappus and the...Ch. 7 - Prob. 11RCCCh. 7 - Prob. 12RCCCh. 7 - Prob. 13RCCCh. 7 - (a) Sketch a direction field for the differential...Ch. 7 - Prob. 44RECh. 7 - Solve the differential equation. 7. 2yey2y=2x+3xCh. 7 - Solve the initial-value problem. 9....Ch. 7 - 41550-9-10RE
9–11. Solve the initial-value...
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- 3.9 (A/B). A beam ABCDE, with A on the left, is 7 m long and is simply supported at Band E. The lengths of the various portions are AB 1-5m, BC = 1-5m, CD = 1 m and DE : 3 m. There is a uniformly distributed load of 15kN/m between B and a point 2m to the right of B and concentrated loads of 20 KN act at 4 and 0 with one of 50 KN at C. (a) Draw the S.F. diagrams and hence determine the position from A at which the S.F. is zero. (b) Determine the value of the B.M. at this point. (c) Sketch the B.M. diagram approximately to scale, quoting the principal values. [3.32 m, 69.8 KNm, 0, 30, 69.1, 68.1, 0 kNm.]arrow_forward4. Verify that V X (aẢ) = (Va) XẢ + aV X Ả where Ả = xyz(x + y + 2) A and a = 3xy + 4zx by carrying out the detailed differentiations.arrow_forward3. For each of the arrow or quiver graphs shown below, determine analytically V°C and V X Č. From these analytical solutions, identify the extrema (+/-) and plot these points on the arrow graph. (a) C = −✰CosxSiny + ŷSinxCosy -π<ׂу<π Ty (b) C = −xSin2y + ŷCos2y x, y<π -π< (c) C = −xCosx + ŷSiny -π< x, y < πarrow_forward
- 7.10 (B/C). A circular flat plate of diameter 305 mm and thickness 6.35 mm is clamped at the edges and subjected to a Uniform lateral pressure of 345 kN/m². Evaluate: (a) the central deflection, (b) the position and magnitude of the maximum radial stress. C6.1 x 10 m; 149.2 MN/m².] 100 200arrow_forward3.15 (B). A beam ABCD is simply supported at B and C with ABCD=2m; BC 4 m. It carries a point load of 60 KN at the free end A, a Uniformly distributed load of 60 KN/m between B and C and an anticlockwise moment of 80 KN m in the plane of the beam applied at the free end D. Sketch and dimension the S.F. and B.M. diagrams, and determine the position and magnitude of the maximum bending moment. CEL.E.] CS.F. 60, 170, 70KN, B.M. 120, +120.1, +80 kNm, 120.1 kNm at 2.83 m to right of 8.7arrow_forward7.1 (A/B). A Uniform I-section beam has flanges 150 mm wide by 8 mm thick and a web 180 mm wide and 8 mm thick. At a certain section there is a shearing force of 120 KN. Draw a diagram to illustrate the distribution of shear stress across the section as a result of bending. What is the maximum shear stress? [86.7 MN/m².arrow_forward
- 1. Let Ả = −2x + 3y+42, B = - - 7x +lý +22, and C = −1x + 2y + 42. Find (a) Ả X B (b) ẢX B°C c) →→ Ả B X C d) ẢB°C e) ẢX B XC.arrow_forward3.13 (B). A beam ABC, 6 m long, is simply-supported at the left-hand end A and at B I'm from the right-hand end C. The beam is of weight 100 N/metre run. (a) Determine the reactions at A and B. (b) Construct to scales of 20 mm = 1 m and 20 mm = 100 N, the shearing-force diagram for the beam, indicating thereon the principal values. (c) Determine the magnitude and position of the maximum bending moment. (You may, if you so wish, deduce the answers from the shearing force diagram without constructing a full or partial bending-moment diagram.) [C.G.] C240 N, 360 N, 288 Nm, 2.4 m from A.]arrow_forward5. Using parentheses make sense of the expression V · VXVV · Å where Ả = Ã(x, y, z). Is the result a vector or a scaler?arrow_forward
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