Calculus: Early Transcendentals

James Stewart

Chapter 8

Further Applications of Integration - all with Video Answers

Educators

Section 3

Applications to Physics and Engineering

Problem 1

An aquarium 5 ft long, 2 ft wide, and 3 ft deep is full of water. Find (a) the hydrostatic pressure on the bottom of the aquarium, (b) the hydrostatic force on the bottom, and (c) the hydrostatic force on one end of the aquarium.

Doruk Isik

Doruk Isik

Numerade Educator

Problem 2

A tank is 8 m long, 4 m wide, 2 m high, and contains kerosene with density $ 820 kg/m^3 $ to a depth of 1.5 m. Find (a) the hydrostatic pressure on the bottom of the tank, (b) the hydrostatic force on the bottom, and (c) the hydrostatic force on one end of the tank.

Doruk Isik

Numerade Educator

Problem 3

A vertical plate is submerged (or partially submerged) in water and has the indicated shape. Explain how to approximate the hydrostatic force against one side of the plate by a Riemann sum.Then express the force as an integral and evaluate it.

Doruk Isik

Numerade Educator

Problem 4

A vertical plate is submerged (or partially submerged) in water and has the indicated shape. Explain how to approximate the hydrostatic force against one side of the plate by a Riemann sum.Then express the force as an integral and evaluate it.

Doruk Isik

Numerade Educator

Problem 5

A vertical plate is submerged (or partially submerged) in water and has the indicated shape. Explain how to approximate the hydrostatic force against one side of the plate by a Riemann sum.Then express the force as an integral and evaluate it.

Doruk Isik

Numerade Educator

Problem 6

A vertical plate is submerged (or partially submerged) in water and has the indicated shape. Explain how to approximate the hydrostatic force against one side of the plate by a Riemann sum.Then express the force as an integral and evaluate it.

Doruk Isik

Numerade Educator

Problem 7

A vertical plate is submerged (or partially submerged) in water and has the indicated shape. Explain how to approximate the hydrostatic force against one side of the plate by a Riemann sum.Then express the force as an integral and evaluate it.

Doruk Isik

Numerade Educator

Problem 8

A vertical plate is submerged (or partially submerged) in water and has the indicated shape. Explain how to approximate the hydrostatic force against one side of the plate by a Riemann sum.Then express the force as an integral and evaluate it.

Doruk Isik

Numerade Educator

Problem 9

A vertical plate is submerged (or partially submerged) in water and has the indicated shape. Explain how to approximate the hydrostatic force against one side of the plate by a Riemann sum.Then express the force as an integral and evaluate it.

Doruk Isik

Numerade Educator

Problem 10

A vertical plate is submerged (or partially submerged) in water and has the indicated shape. Explain how to approximate the hydrostatic force against one side of the plate by a Riemann sum.Then express the force as an integral and evaluate it.

Doruk Isik

Numerade Educator

Problem 11

A vertical plate is submerged (or partially submerged) in water and has the indicated shape. Explain how to approximate the hydrostatic force against one side of the plate by a Riemann sum.Then express the force as an integral and evaluate it.

Doruk Isik

Numerade Educator

Problem 12

A milk truck carries milk with density $ 64.6 lb/ft^3 $ in a horizontal cylindrical tank with diameter 6 ft.
(a) Find the force exerted by the milk on one end of the tank when the tank is full.
(b) What if the tank is half full?

Doruk Isik

Numerade Educator

Problem 13

A trough is filled with a liquid of density $ 840 kg/m^3 $. The ends of the trough are equilateral triangles with sides 8 m long and vertex at the bottom. Find the hydrostatic force on one end of the trough.

Doruk Isik

Numerade Educator

Problem 14

A vertical dam has a semicircular gate as shown in the figure. Find the hydrostatic force against the gate.

Doruk Isik

Numerade Educator

Problem 15

A cube with 20-cm-long sides is sitting on the bottom of an aquarium in which the water is one meter deep. Find the hydrostatic force on (a) the top of the cube and (b) one of the sides of the cube.

Doruk Isik

Numerade Educator

Problem 16

A dam is inclined at an angle of $ 30^\circ $ from the vertical and has the shape of an isosceles trapezoid 100 ft wide at the top and 50 ft wide at the bottom and with a slant height of 70 ft. Find the hydrostatic force on the dam when it is full of water.

Doruk Isik

Numerade Educator

Problem 17

A swimming pool is 20 ft wide and 40 ft long and its bottom is an inclined plane, the shallow end having a depth of 3 ft and the deep end, 9 ft. If the pool is full of water, find the hydrostatic force on (a) the shallow end, (b) the deep end, (c) one of the sides, and (d) the bottom of the pool.

Sam Low

Numerade Educator

Problem 18

Suppose that a plate is immersed vertically in a fluid with density $ \rho $ and the width of the plate is $ w(x) $ at a depth of $ x $ meters beneath the surface of the fluid. If the top of the plate is at depth $ a $ and the bottom is at depth $ b $, show that the hydrostatic force on one side of the plate is
$$ F = \int_a^b \rho gxw(x)\ dx $$

Doruk Isik

Numerade Educator

Problem 19

A metal plate was found submerged vertically in seawater, which has density $ 64 lb/ft^3 $. Measurements of the width of the plate were taken at the indicated depths. Use Simpson's Rule to estimate the force of the water against the plate.

Doruk Isik

Numerade Educator

Problem 20

(a) Use the formula of Exercise 18 to show that
$$ F = (\rho g \bar{x}) A $$
where $ \bar{x} $ is the x-coordinate of the centroid of the plate and $ A $ is its area. This equation shows that the hydrostatic force against a vertical plane region is the same as if the region were horizontal at the depth of the centroid of the region.
(b) Use the result of part (a) to give another solution to Exercise 10.

Doruk Isik

Numerade Educator

Problem 21

Point-masses $ m_i $ are located on the x-axis as shown. Find the moment $ M $ of the system about the origin and the center of mass $ \bar{x} $.

Doruk Isik

Numerade Educator

Problem 22

Point-masses $ m_i $ are located on the x-axis as shown. Find the moment $ M $ of the system about the origin and the center of mass $ \bar{x} $.

Doruk Isik

Numerade Educator

Problem 23

The masses $ m_i $ are located at the points $ P_i $. Find the moments $ M_x $ and $ M_y $ and the center of mass of the system.

$ m_1 = 4 $ , $ m_2 = 2 $ , $ m_3 = 4 $ ;
$ P_1 (2, -3) $ , $ P_2 (-3, 1) $ , $ P_3 (3, 5) $

Doruk Isik

Numerade Educator

Problem 24

The masses $ m_i $ are located at the points $ P_i $. Find the moments $ M_x $ and $ M_y $ and the center of mass of the system.

$ m_1 = 5 $ , $ m_2 = 4 $ , $ m_3 = 3 $ , $ m_4 = 6 $ ;
$ P_1 (-4, 2) $ , $ P_2 (0, 5) $ , $ P_3 (3, 2) $ , $ P_4 (1, -2) $

Doruk Isik

Numerade Educator

Problem 25

Sketch the region bounded by the curves, and visually estimate the location of the centroid. Then find the exact coordinates of the centroid.

$ y = 2x $ , $ y = 0 $ , $ x = 1 $

Doruk Isik

Numerade Educator

Problem 26

Sketch the region bounded by the curves, and visually estimate the location of the centroid. Then find the exact coordinates of the centroid.

$ y = \sqrt{x} $ , $ y = 0 $ , $ x = 4 $

Doruk Isik

Numerade Educator

Problem 27

Sketch the region bounded by the curves, and visually estimate the location of the centroid. Then find the exact coordinates of the centroid.

$ y = e^x $ , $ y = 0 $ , $ x = 0 $ , $ x = 1 $

Doruk Isik

Numerade Educator

Problem 28

Sketch the region bounded by the curves, and visually estimate the location of the centroid. Then find the exact coordinates of the centroid.

$ y = \sin x $ , $ y = 0 $ , $ 0 \le x \le \pi $

Doruk Isik

Numerade Educator

Problem 29

Find the centroid of the region bounded by the given curves.

$ y = x^2 $ , $ x = y^2 $

Doruk Isik

Numerade Educator

Problem 30

Find the centroid of the region bounded by the given curves.

$ y = 2 - x^2 $ , $ y = x $

Doruk Isik

Numerade Educator

Problem 31

Find the centroid of the region bounded by the given curves.

$ y = \sin x $ , $ y = \cos x $ , $ x = 0 $ , $ x = \frac{\pi}{4} $

Doruk Isik

Numerade Educator

Problem 32

Find the centroid of the region bounded by the given curves.

$ y = x^3 $ , $ x + y = 2 $ , $ y = 0 $

Doruk Isik

Numerade Educator

Problem 33

Find the centroid of the region bounded by the given curves.

$ x + y = 2 $ , $ x = y^2 $

Doruk Isik

Numerade Educator

Problem 34

Calculate the moments $ M_x $ and $ M_y $ and the center of mass of a lamina with the given density and shape.

$ \rho = 4 $

Doruk Isik

Numerade Educator

Problem 35

Calculate the moments $ M_x $ and $ M_y $ and the center of mass of a lamina with the given density and shape.

$ \rho = 6 $

Doruk Isik

Numerade Educator

Problem 36

Use Simpson's Rule to estimate the centroid of the region shown.

Doruk Isik

Numerade Educator

Problem 37

Find the centroid of the region bounded by the curves $ y = x^3 - x $ and $ y = x^2 - 1 $. Sketch the region and plot the centroid to see if your answer is reasonable.

Doruk Isik

Numerade Educator

Problem 38

Use a graph to find approximate x-coordinates of the points of intersection of the curves $ y = e^x $ and $ y = 2 - x^2 $. Then find (approximately) the centroid of the region bounded by these curves.

Doruk Isik

Numerade Educator

Problem 39

Prove that the centroid of any triangle is located at the point of intersection of the medians. [Hints: Place the axes so that the vertices are $ (a, 0) $, $ (0, b) $ , and $ (c, 0) $. Recall that a median is a line segment from a vertex to the midpoint of the opposite side. Recall also that the medians intersect at a point two-thirds of the way from each vertex (along the median) to the opposite side.]

Doruk Isik

Numerade Educator

Problem 40

Find the centroid of the region shown, not by integration, but by locating the centroids of the rectangles and triangles (from Exercise 39) and using additivity of moments.

Doruk Isik

Numerade Educator

Problem 41

Find the centroid of the region shown, not by integration, but by locating the centroids of the rectangles and triangles (from Exercise 39) and using additivity of moments.

Doruk Isik

Numerade Educator

Problem 42

A rectangle $ \Re $ with sides $ a $ and $ b $ is divided into two parts $ \Re_1 $ and $ \Re_2 $ by an arc of a parabola that has its vertex at one corner of $ \Re $ and passes through the opposite corner. Find the centroids of both $ \Re_1 $ and $ \Re_2 $.

Doruk Isik

Numerade Educator

Problem 43

If $ \bar{x} $ is the x-coordinate of the centroid of the region that lies under the graph of a continuous function $ f $, where $ a \le x \le b $, show that
$$ \int_a^b (cx + d) f(x) dx = (c \bar{x} + d) \int_a^b f(x) dx $$

Doruk Isik

Numerade Educator

Problem 44

Use the Theorem of Pappus to find the volume of the given solid.
A sphere of radius $ r $ (Use Example 4.)

Doruk Isik

Numerade Educator

Problem 45

Use the Theorem of Pappus to find the volume of the given solid.
A cone with height $ h $ and base radius $ r $.

Doruk Isik

Numerade Educator

Problem 46

Use the Theorem of Pappus to find the volume of the given solid.
The solid obtained by rotating the triangle with vertices $ (2, 3) $, $ (2, 5) $, and $ (5, 4) $ about the x-axis.

Doruk Isik

Numerade Educator

Problem 47

The centroid of a $ curve $ can be found by a process similar to the one we used for finding the centroid of a region. If $ C $ is a curve with length $ L $, then the centroid is $ (\bar{x}, \bar{y}) $ where $ \bar{x} = (\frac{1}{L}) \int x\ ds $ and $ \bar{y} = (\frac{1}{L}) \int y\ ds $. Here we assign appropriate limits of integration, and $ ds $ is as defined in Sections 8.1 and 8.2. ( The centroid often doesn't lie on the curve itself. If the curve were made of wire and placed on a weightless board, the centroid would be the balance point on the board.) Find the centroid of the quarter-circle $ y = \sqrt{16 - x^2} $, $ 0 \le x \le 4 $.

Doruk Isik

Numerade Educator

Problem 48

The Second Theorem of Pappus is in the same spirit as Pappus's Theorem on page 565, but for surface area rather than volume: Let $ C $ be a curve that lies entirely on one side of a line $ l $ in the plane. If $ C $ is rotated about $ l $, then the area of the resulting surface is the product of the arc length of $ C $ and the distance traveled by the centroid of $ C $ (see Exercise 47).

(a) Prove the Second Theorem of Pappus for the case where $ C $ is given by $ y = f(x), f(x) \ge 0 $ and $ C $. is rotated about the x-axis.
(b) Use the Second Theorem of Pappus to compute the surface area of the half-sphere obtained by rotating the curve from Exercise 47 about the x-axis. Does your answer agree with the one given by geometric formulas?

Doruk Isik

Numerade Educator

Problem 49

Use the Second Theorem of Pappus described in Exercise 48 to find the surface area of the torus in Example 7.

Doruk Isik

Numerade Educator

Problem 50

Let $ \Re $ be the region that lies between the curves
$ y = x^m $ $ y = x^n $ $ 0 \le x \le 1 $
where $ m $ and $ n $ are integers with $ 0 \le n < m $
(a) Sketch the region $ \Re $.
(b) Find the coordinates of the centroid of $ \Re $.
(c) Try to find values of $ m $ and $ n $ such that the centroid lies outside $ \Re $.

Doruk Isik

Numerade Educator

Problem 51

Prove Formulas 9.

Sam Low

Numerade Educator