Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.

$$y=2-\frac{1}{2} x, y=0, x=1, x=2 ; \text { about the } x-axis$$

Rebecca P.

Numerade Educator

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.

$$y=1-x^{2}, y=0 ; \quad \text { about the } x-axis$$

Rebecca P.

Numerade Educator

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.

$$x=2 \sqrt{y}, x=0, y=9 ; \quad \text { about the } y-axis$$

Rebecca P.

Numerade Educator

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.

$$y=\ln x, y=1, y=2, x=0 ; \quad \text { about the } y-axis$$

Rebecca P.

Numerade Educator

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.

$$y=x^{3}, y=x, x \geqslant 0 ; \quad \text { about the } x-axis$$

Rebecca P.

Numerade Educator

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.

$$y=\frac{1}{4} x^{2}, y=5-x^{2} ; \quad \text { about the } x-axis$$

Rebecca P.

Numerade Educator

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.

$$y^{2}=x, x=2 y ; \quad \text { about the } y-axis$$

Rebecca P.

Numerade Educator

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.

$$y=\frac{1}{4} x^{2}, x=2, y=0 ; \quad \text { about the } y-axis$$

Rebecca P.

Numerade Educator

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.

$$y=x, y=\sqrt{x} ; \quad \text { about } y=1$$

Rebecca P.

Numerade Educator

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.

$$y=e^{-x}, y=1, x=2 ; \quad \text { about } y=2$$

Rebecca P.

Numerade Educator

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.

$$y=1+\sec x, y=3 ; \quad \text { about } y=1$$

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Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.

$$y=x, y=\sqrt{x} ; \text { about } x=2$$

Rebecca P.

Numerade Educator

The region enclosed by the given curves is rotated about the specified line. Find the volume of the resulting solid.

$$y=1 / x, x=1, x=2, y=0 ; \quad \text { about the } x-axis$$

Rebecca P.

Numerade Educator

The region enclosed by the given curves is rotated about the specified line. Find the volume of the resulting solid.

$$x=2 y-y^{2}, x=0 ; \quad \text { about the } y-axis$$

Rebecca P.

Numerade Educator

The region enclosed by the given curves is rotated about the specified line. Find the volume of the resulting solid.

$$x-y=1, y=x^{2}-4 x+3 ; \quad \text { about } y=3$$

Rebecca P.

Numerade Educator

The region enclosed by the given curves is rotated about the specified line. Find the volume of the resulting solid.

$$x=y^{2}, x=1 ; \quad \text { about } x=1$$

Rebecca P.

Numerade Educator

The region enclosed by the given curves is rotated about the specified line. Find the volume of the resulting solid.

$$y=x^{3}, y=\sqrt{x} ; \quad \text { about } x=1$$

Rebecca P.

Numerade Educator

The region enclosed by the given curves is rotated about the specified line. Find the volume of the resulting solid.

$$y=x^{3}, y=\sqrt{x} ; \quad \text { about } y=1$$

Rebecca P.

Numerade Educator

Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Then use your calculator to evaluate the integral correct to five decimal places.

$$y=e^{-x^{2}}, y=0, x=-1, x=1$$

(a) About the $x$ -axis (b) About $y=-1$

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Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Then use your calculator to evaluate the integral correct to five decimal places.

$$y=0, y=\cos ^{2} x,-\pi / 2 \leqslant x \leqslant \pi / 2$$

(a) About the $x$ -axis (b) About y $=1$

Victoria D.

Numerade Educator

Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Then use your calculator to evaluate the integral correct to five decimal places.

$$x^{2}+4 y^{2}=4$$

(a) About $y=2$ (b) About $x=2$

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Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Then use your calculator to evaluate the integral correct to five decimal places.

$$y=x^{2}, x^{2}+y^{2}=1, y \geqslant 0$$

(a) About the $x$ -axis (b) About the $y$ -axis

Rebecca P.

Numerade Educator

Use a graph to find approximate $x$ -coordinates of the points of intersection of the given curves. Then use your calculator to find (approximately) the volume of the solid obtained

by rotating about the $x$ -axis the region bounded by these curves.

$$y=2+x^{2} \cos x, \quad y=x^{4}+x+1$$

Victoria D.

Numerade Educator

Use a graph to find approximate $x$ -coordinates of the points of intersection of the given curves. Then use your calculator to find (approximately) the volume of the solid obtained

by rotating about the $x$ -axis the region bounded by these curves.

$$y=3 \sin \left(x^{2}\right), \quad y=e^{x / 2}+e^{-2 x}$$

Victoria D.

Numerade Educator

Use a computer algebra system to find the exact volume of the solid obtained by rotating the region bounded by the given curves about the specified line.

$$y=\sin ^{2} x, y=0,0 \leqslant x \leqslant \pi ; \quad \text { about } y=-1$$

Victoria D.

Numerade Educator

Use a computer algebra system to find the exact volume of the solid obtained by rotating the region bounded by the given curves about the specified line.

$$y=x, y=x e^{1-x / 2} ; \quad \text { about } y=3$$

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Each integral represents the volume of a solid. Describe the solid.

$$ \text { (a) }\pi \int_{0}^{\pi / 2} \cos ^{2} x d x \quad \text { (b) } \pi \int_{0}^{1}\left(y^{4}-y^{8}\right) d y$$

Nick J.

Numerade Educator

Each integral represents the volume of a solid.

Describe the solid.

$$

\text { (a) }\pi \int_{2}^{5} y d y \quad \text { (b) } \pi \int_{0}^{\pi / 2}\left[(1+\cos x)^{2}-1^{2}\right] d x

$$

Victoria D.

Numerade Educator

A CAT scan produces equally spaced cross-sectional views

of a human organ that provide information about the organ

otherwise obtained only by surgery. Suppose that a CAT

scan of a human liver shows cross-scctions spaced 1.5 $\mathrm{cm}$

apart. The liver is 15 $\mathrm{cm}$ long and the cross-sectional areas,

in square centimeters, are $0,18,58,79,94,106,117,128,$

$63,39,$ and $0 .$ Use the Midpoint Rulc to estimate the volume of the liver.

Rebecca P.

Numerade Educator

A log 10 $\mathrm{m}$ long is cut at 1 -meter intervals and its cross-

sectional areas $A$ (at a distance $x$ from the end of the log)

are listed in the table. Use the Midpoint Rule with $n=5$

to estimate the volume of the log.

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Find the volume of the described solid $S .$

A right circular cone with height $h$ and base radius $r$

Rebecca P.

Numerade Educator

Find the volume of the described solid $S .$

A frustum of a right circular cone with height $h$ , lower

base radius $R,$ and top radius $r$

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Find the volume of the described solid $S .$

A cap of a sphere with radius $r$ and height $h$

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Find the volume of the described solid $S .$

A frustum of a pyramid with square base of side $b,$ square

top of side $a,$ and height $h$

What happens if $a=b ?$ What happens if $a=0 ?$

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Find the volume of the described solid $S .$

A pyramid with height $h$ and rectangular base with dimensions $b$ and 2$b$

Rebecca P.

Numerade Educator

Find the volume of the described solid $S .$

A pyramid with height $h$ and base an cquilateral triangle

with side $a($ a tetrahedron $)$

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Find the volume of the described solid $S .$

A tetrahedron with three mutually perpendicular faces and

three mutually perpendicular cdges with lengths 3 $\mathrm{cm}$ ,

$4 \mathrm{cm},$ and 5 $\mathrm{cm}$

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Find the volume of the described solid $S .$

The base of $S$ is a circular disk with radius $r .$ Parallel cross-sections perpendicular to the base are squares.

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Find the volume of the described solid $S .$

The base of $S$ is an elliptical region with boundary curve

$9 x^{2}+4 y^{2}=36 .$ Cross-sections perpendicular to the $x$ -axis

are isosceles right triangles with hypotenuse in the base.

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Find the volume of the described solid $S .$

The base of $S$ is the triangular region with vertices $(0,0),$

$(1,0),$ and $(0,1)$ . Cross-sections perpendicular to the $y$ -axis

are equilateral triangles.

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Find the volume of the described solid $S .$

The base of $S$ is the same base as in Exercise 40 , but cross-sections perpendicular to the $x$ -axis are squares.

Nick J.

Numerade Educator

Find the volume of the described solid $S .$

The base of $S$ is the region enclosed by the parabola

$y=1-x^{2}$ and the $x$ -axis. Cross-sections perpendicular to

the $y$ -axis are squares.

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Find the volume of the described solid $S .$

The base of $S$ is the same base as in Exercise $42,$ but cross-sections perpendicular to the $x$ -axis are isosceles triangles

with height equal to the base.

Nick J.

Numerade Educator

The base of $S$ is a circular disk with radius $r .$ Parallel cross-sections perpendicular to the base are isosceles triangles

with height $h$ and unequal side in the base.

(a) Set up an integral for the volume of $S .$

(b) By interpreting the integral as an area, find the volume

of $S .$

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Some of the pioneers of calculus, such as Kepler and Newton, were inspired by the problem of finding the volumes of wine barrels. (In fact Kepler published a book Stereometria doliorum in 1715 devoted to methods for finding the

volumes of barrels.) They often approximated the shape of

the sides by parabolas.

(a) A barrel with height $h$ and maximum radius $R$ is constructed by rotating about the $x$ -axis the parabola

$y=R-c x^{2},-h / 2 \leqslant x \leqslant h / 2,$ where $c$ is a positive

constant. Show that the radius of each end of the barrel

is $r=R-d,$ where $d=c h^{2} / 4$

(b) Show that the volume enclosed by the barrel is

$$

V=\frac{1}{3} \pi h\left(2 R^{2}+r^{2}-\frac{2}{5} d^{2}\right)

$$

Victoria D.

Numerade Educator

(a) A model for the shape of a bird's egg is obtained by

rotating about the $x$ -axis the region under the graph of

$$

f(x)=\left(a x^{3}+b x^{2}+c x+d\right) \sqrt{1-x^{2}}

$$

Use a CAS to find the volume of such an egg.

(b) For a Red-throated Loon, $a=-0.06, b=0.04$

$c=0.1,$ and $d=0.54 .$ Graph $f$ and find the volume

of an egg of this bird.

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(a) Set up an integral for the volume of a solid torus (the

donut-shaped solid shown in the figure) with radii $r$

and $R .$

(b) By interpreting the integral as an area, find the volume

of the torus.

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A wedge is cut out of a circular cylinder of radius 4 by two

planes. One plane is perpendicular to the axis of the cylinder. The other intersects the first at an angle of $30^{\circ}$ along a

diameter of the cylinder. Find the volume of the wedge.

Paul A.

California State Polytechnic University, Pomona

(a) Cavalieri's Principle states that if a family of parallel

planes gives equal cross-sectional areas for two solids

$S_{1}$ and $S_{2},$ then the volumes of $S_{1}$ and $S_{2}$ are equal.

Prove this principle.

(b) Use Cavalieri's Principle to find the volume of the

oblique cylinder shown in the figure.

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Find the volume common to two circular cylinders, each

with radius $r,$ if the axes of the cylinders intersect at right

angles.

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Find the volume common to two spheres, each with radius $r$

if the center of each sphere lies on the surface of the other

sphere.

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A bowl is shaped like a hemisphere with diameter 30 $\mathrm{cm} . \mathrm{A}$

ball with diameter 10 $\mathrm{cm}$ is placed in the bowl and water is

poured into the bowl to a depth of $h$ centimeters. Find the

volume of water in the bowl.

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A hole of radius $r$ is bored through a cylinder of radius

$R>r$ at right angles to the axis of the cylinder. Set up, but

do not evaluate, an integral for the volume cut out.

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A hole of radius $r$ is bored through the center of a sphere of

radius $R > r .$ Find the volume of the remaining portion of

the sphere.

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