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A tetrahedron with a base side of 4 units, as seen here.
$\frac{32}{3\sqrt{2}}$ units ^{3}
A pyramid with height 5 units, and an isosceles triangular base with lengths of 6 units and 8 units, as seen here.
A cone of radius $r$ and height $h$ has a smaller cone of radius $r\text{/}2$ and height $h\text{/}2$ removed from the top, as seen here. The resulting solid is called a frustum .
$\frac{7\pi}{12}h{r}^{2}$ units ^{3}
For the following exercises, draw an outline of the solid and find the volume using the slicing method.
The base is a circle of radius $a.$ The slices perpendicular to the base are squares.
The base is a triangle with vertices $\left(0,0\right),\left(1,0\right),$ and $\left(0,1\right).$ Slices perpendicular to the xy -plane are semicircles.
$\frac{\pi}{24}$ units
^{3}
The base is the region under the parabola $y=1-{x}^{2}$ in the first quadrant. Slices perpendicular to the xy -plane are squares.
The base is the region under the parabola $y=1-{x}^{2}$ and above the $x\text{-axis}\text{.}$ Slices perpendicular to the $y\text{-axis}$ are squares.
$2$ units
^{3}
The base is the region enclosed by $y={x}^{2}$ and $y=9.$ Slices perpendicular to the x -axis are right isosceles triangles.
The base is the area between $y=x$ and $y={x}^{2}.$ Slices perpendicular to the x -axis are semicircles.
$\frac{\pi}{240}$ units
^{3}
For the following exercises, draw the region bounded by the curves. Then, use the disk method to find the volume when the region is rotated around the x -axis.
$x+y=8,x=0,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}y=0$
$y=2{x}^{2},x=0,x=4,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}y=0$
$\frac{4096\pi}{5}$ units
^{3}
$y={e}^{x}+1,x=0,x=1,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}y=0$
$y={x}^{4},x=0,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}y=1$
$\frac{8\pi}{9}$ units
^{3}
$y=\sqrt{x},x=0,x=4,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}y=0$
$y=\text{sin}\phantom{\rule{0.2em}{0ex}}x,y=\text{cos}\phantom{\rule{0.2em}{0ex}}x,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}x=0$
$\frac{\pi}{2}$ units
^{3}
$y=\frac{1}{x},x=2,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}y=3$
${x}^{2}-{y}^{2}=9\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}x+y=9,y=0\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}x=0$
$207\pi $ units
^{3}
For the following exercises, draw the region bounded by the curves. Then, find the volume when the region is rotated around the y -axis.
$y=4-\frac{1}{2}x,x=0,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}y=0$
$y=2{x}^{3},x=0,x=1,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}y=0$
$\frac{4\pi}{5}$ units
^{3}
$y=3{x}^{2},x=0,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}y=3$
$y=\sqrt{4-{x}^{2}},y=0,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}x=0$
$\frac{16\pi}{3}$ units
^{3}
$y=\frac{1}{\sqrt{x+1}},x=0,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}x=3$
$x=\text{sec}\left(y\right)\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}y=\frac{\pi}{4},y=0\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}x=0$
$\pi $ units
^{3}
$y=\frac{1}{x+1},x=0,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}x=2$
$y=4-x,y=x,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}x=0$
$\frac{16\pi}{3}$ units
^{3}
For the following exercises, draw the region bounded by the curves. Then, find the volume when the region is rotated around the x -axis.
$y=x+2,y=x+6,x=0,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}x=5$
$y={x}^{2}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}y=x+2$
$\frac{72\pi}{5}$ units
^{3}
${x}^{2}={y}^{3}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}{x}^{3}={y}^{2}$
$y=4-{x}^{2}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}y=2-x$
$\frac{108\pi}{5}$ units
^{3}
[T] $y=\text{cos}\phantom{\rule{0.2em}{0ex}}x,y={e}^{\text{\u2212}x},x=0,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}x=1.2927$
$y=\sqrt{x}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}y={x}^{2}$
$\frac{3\pi}{10}$ units
^{3}
$y=\text{sin}\phantom{\rule{0.2em}{0ex}}x\text{,}\phantom{\rule{0.2em}{0ex}}y=5\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}x,x=0\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}x=\pi $
$y=\sqrt{1+{x}^{2}}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}y=\sqrt{4-{x}^{2}}$
$2\sqrt{6}\pi $ units
^{3}
For the following exercises, draw the region bounded by the curves. Then, use the washer method to find the volume when the region is revolved around the y -axis.
$y=\sqrt{x},x=4,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}y=0$
$y=x+2,y=2x-1,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}x=0$
$9\pi $ units
^{3}
$y=\sqrt[3]{x}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}y={x}^{3}$
$x={e}^{2y},x={y}^{2},y=0,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}y=\text{ln}\left(2\right)$
$\frac{\pi}{20}\left(75-4\phantom{\rule{0.2em}{0ex}}{\text{ln}}^{5}\left(2\right)\right)$ units
^{3}
$x=\sqrt{9-{y}^{2}},x={e}^{\text{\u2212}y},y=0,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}y=3$
Yogurt containers can be shaped like frustums. Rotate the line $y=\frac{1}{m}x$ around the y -axis to find the volume between $y=a\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}y=b.$
$\frac{{m}^{2}\pi}{3}\left({b}^{3}-{a}^{3}\right)$ units ^{3}
Rotate the ellipse $\left({x}^{2}\text{/}{a}^{2}\right)+\left({y}^{2}\text{/}{b}^{2}\right)=1$ around the x -axis to approximate the volume of a football, as seen here.
Rotate the ellipse $\left({x}^{2}\text{/}{a}^{2}\right)+\left({y}^{2}\text{/}{b}^{2}\right)=1$ around the y -axis to approximate the volume of a football.
$\frac{4{a}^{2}b\pi}{3}$ units ^{3}
A better approximation of the volume of a football is given by the solid that comes from rotating $y=\text{sin}\phantom{\rule{0.2em}{0ex}}x$ around the x -axis from $x=0$ to $x=\pi .$ What is the volume of this football approximation, as seen here?
What is the volume of the Bundt cake that comes from rotating $y=\text{sin}\phantom{\rule{0.2em}{0ex}}x$ around the y -axis from $x=0$ to $x=\pi ?$
$2{\pi}^{2}$ units ^{3}
For the following exercises, find the volume of the solid described.
The base is the region between $y=x$ and $y={x}^{2}.$ Slices perpendicular to the x -axis are semicircles.
The base is the region enclosed by the generic ellipse $\left({x}^{2}\text{/}{a}^{2}\right)+\left({y}^{2}\text{/}{b}^{2}\right)=1.$ Slices perpendicular to the x -axis are semicircles.
$\frac{2a{b}^{2}\pi}{3}$ units ^{3}
Bore a hole of radius $a$ down the axis of a right cone and through the base of radius $b,$ as seen here.
Find the volume common to two spheres of radius $r$ with centers that are $2h$ apart, as shown here.
$\frac{\pi}{12}{\left(r+h\right)}^{2}\left(6r-h\right)$ units ^{3}
Find the volume of a spherical cap of height $h$ and radius $r$ where $h<r,$ as seen here.
Find the volume of a sphere of radius $R$ with a cap of height $h$ removed from the top, as seen here.
$\frac{\pi}{3}\left(h+R\right){\left(h-2R\right)}^{2}$ units ^{3}
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