Calculus

Deborah Hughes-Hallett, Andrew M. Gleason, William G. McCallum

Chapter 8

Using the Definite Integral - all with Video Answers

Educators

Section 1

Areas and Volumes

Problem 1

Write a Riemann sum and then a definite integral representing the area of the region, using the strip shown. Evaluate the integral exactly.
(FIGURE CAN'T COPY)

Lucas Finney

Lucas Finney

Numerade Educator

Problem 2

Write a Riemann sum and then a definite integral representing the area of the region, using the strip shown. Evaluate the integral exactly.
(FIGURE CAN'T COPY)

Lucas Finney

Numerade Educator

Problem 3

Write a Riemann sum and then a definite integral representing the area of the region, using the strip shown. Evaluate the integral exactly.
(FIGURE CAN'T COPY)

Lucas Finney

Numerade Educator

Problem 4

Write a Riemann sum and then a definite integral representing the area of the region, using the strip shown. Evaluate the integral exactly.
(FIGURE CAN'T COPY)

Lucas Finney

Numerade Educator

Problem 5

Write a Riemann sum and then a definite integral representing the area of the region, using the strip shown. Evaluate the integral exactly.
(FIGURE CAN'T COPY)

Lucas Finney

Numerade Educator

Problem 6

Write a Riemann sum and then a definite integral representing the area of the region, using the strip shown. Evaluate the integral exactly.
(FIGURE CAN'T COPY)

Lucas Finney

Numerade Educator

Problem 7

Write a Riemann sum and then a definite integral representing the area of the region, using the strip shown. Evaluate the integral exactly.
(FIGURE CAN'T COPY)

Lucas Finney

Numerade Educator

Problem 8

Write a Riemann sum and then a definite integral representing the area of the region, using the strip shown. Evaluate the integral exactly.
(FIGURE CAN'T COPY)

Lucas Finney

Numerade Educator

Problem 9

Write a Riemann sum and then a definite integral representing the volume of the region, using the slice shown. Evaluate the integral exactly. (Regions are parts of cones, cylinders, spheres, and pyramids.)
(FIGURE CAN'T COPY)

Lucas Finney

Numerade Educator

Problem 10

Write a Riemann sum and then a definite integral representing the volume of the region, using the slice shown. Evaluate the integral exactly. (Regions are parts of cones, cylinders, spheres, and pyramids.)
(FIGURE CAN'T COPY)

Lucas Finney

Numerade Educator

Problem 11

Write a Riemann sum and then a definite integral representing the volume of the region, using the slice shown. Evaluate the integral exactly. (Regions are parts of cones, cylinders, spheres, and pyramids.)
(FIGURE CAN'T COPY)

Lucas Finney

Numerade Educator

Problem 12

Write a Riemann sum and then a definite integral representing the volume of the region, using the slice shown. Evaluate the integral exactly. (Regions are parts of cones, cylinders, spheres, and pyramids.)
(FIGURE CAN'T COPY)

Lucas Finney

Numerade Educator

Problem 13

Write a Riemann sum and then a definite integral representing the volume of the region, using the slice shown. Evaluate the integral exactly. (Regions are parts of cones, cylinders, spheres, and pyramids.)
(FIGURE CAN'T COPY)

Lucas Finney

Numerade Educator

Problem 14

Write a Riemann sum and then a definite integral representing the volume of the region, using the slice shown. Evaluate the integral exactly. (Regions are parts of cones, cylinders, spheres, and pyramids.)
(FIGURE CAN'T COPY)

Lucas Finney

Numerade Educator

Problem 15

Represent the area of either a triangle or part of a circle, and the variable of integration measures a distance. In each case, say which shape is represented, and give the radius of the circle or the base and height of the triangle. Make a sketch to support your answer showing the variable and all other relevant quantities.
$$\int_{0}^{1} 3 x d x$$

Gregory Higby

Numerade Educator

Problem 16

Represent the area of either a triangle or part of a circle, and the variable of integration measures a distance. In each case, say which shape is represented, and give the radius of the circle or the base and height of the triangle. Make a sketch to support your answer showing the variable and all other relevant quantities.
$$\int_{-9}^{9} \sqrt{81-x^{2}} d x$$

Gregory Higby

Numerade Educator

Problem 17

Represent the area of either a triangle or part of a circle, and the variable of integration measures a distance. In each case, say which shape is represented, and give the radius of the circle or the base and height of the triangle. Make a sketch to support your answer showing the variable and all other relevant quantities.
$$\int_{0}^{\sqrt{15}} \sqrt{15-h^{2}} d h$$

Gregory Higby

Numerade Educator

Problem 18

Represent the area of either a triangle or part of a circle, and the variable of integration measures a distance. In each case, say which shape is represented, and give the radius of the circle or the base and height of the triangle. Make a sketch to support your answer showing the variable and all other relevant quantities.
$$\int_{0}^{7} 5\left(1-\frac{h}{7}\right) d h$$

Gregory Higby

Numerade Educator

Problem 19

The integrals in Problems $19-22$ represent the volume of either a hemisphere or a cone, and the variable of integration measures a length. In each case, say which shape is represented, and give the radius of the hemisphere or the radius and height of the cone. Make a sketch to support your answer showing the variable and all other relevant quantities.
$$\int_{0}^{12} \pi\left(144-h^{2}\right) d h$$

Gregory Higby

Numerade Educator

Problem 20

Represent the volume of either a hemisphere or a cone, and the variable of integration measures a length. In each case, say which shape is represented, and give the radius of the hemisphere or the radius and height of the cone. Make a sketch to support your answer showing the variable and all other relevant quantities.
$$\int_{0}^{12} \pi(x / 3)^{2} d x$$

Gregory Higby

Numerade Educator

Problem 21

Represent the volume of either a hemisphere or a cone, and the variable of integration measures a length. In each case, say which shape is represented, and give the radius of the hemisphere or the radius and height of the cone. Make a sketch to support your answer showing the variable and all other relevant quantities.
$$\int_{0}^{6} \pi(3-y / 2)^{2} d y$$

Gregory Higby

Numerade Educator

Problem 22

Represent the volume of either a hemisphere or a cone, and the variable of integration measures a length. In each case, say which shape is represented, and give the radius of the hemisphere or the radius and height of the cone. Make a sketch to support your answer showing the variable and all other relevant quantities.
$$\int_{0}^{2} \pi\left(2^{2}-(2-y)^{2}\right) d y$$

Gregory Higby

Numerade Educator

Problem 23

Find, by slicing, the volume of a cone whose height is $3 \mathrm{cm}$ and whose base radius is $1 \mathrm{cm} .$ Slice the cone as shown in Figure 8.4 on page 393.

Amrita Bhasin

Numerade Educator

Problem 24

Find the volume of a sphere of radius $r$ by slicing.

Israel Hernandez

Israel Hernandez

Numerade Educator

Problem 25

Find, by slicing, a formula for the volume of a cone of height $h$ and base radius $r.$

Israel Hernandez

Israel Hernandez

Numerade Educator

Problem 26

Figure 8.13 shows a solid with both rectangular and triangular cross sections.(a) Slice the solid parallel to the triangular faces. Sketch one slice and calculate its volume in terms of $x,$ the distance of the slice from one end. Then write and evaluate an integral giving the volume of the solid. (b) Repeat part (a) for horizontal slices. Instead of $x$, use $h,$ the distance of a slice from the top. (FIGURE CAN'T COPY)

Uma Kumari

Numerade Educator

Problem 27

A rectangular lake is $150 \mathrm{km}$ long and $3 \mathrm{km}$ wide. The vertical cross-section through the lake in Figure 8.14 shows that the lake is $0.2 \mathrm{km}$ deep at the center. (These are the approximate dimensions of Lake Mead, the largest reservoir in the US, which provides water to California, Nevada, and Arizona.) Set up and evaluate a definite integral giving the total volume of the lake. (FIGURE CAN'T COPY)

Bobby Barnes

University of North Texas

Problem 28

A dam has a rectangular base 1400 meters long and 160 meters wide. Its cross-section is shown in Figure 8.15 (The Grand Coulee Dam in Washington state is roughly this size.) By slicing horizontally, set up and evaluate a definite integral giving the volume of material used to build this dam. (FIGURE CAN'T COPY)

Tanishq Gupta

Numerade Educator