6.2 Determining volumes by slicing  (Page 7/12)

A tetrahedron with a base side of 4 units, as seen here.

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 .

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.

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.

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.

$x+y=8,x=0,\text{and}y=0$

$y=2x^2,x=0,x=4,\text{and}y=0$

$y=e^x+1,x=0,x=1,\text{and}y=0$

$y=x^4,x=0,\text{and}y=1$

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

$y=\text{sin}x,y=\text{cos}x,\text{and}x=0$

$y=\frac{1}{x},x=2,\text{and}y=3$

$x^2-y^2=9\text{and}x+y=9,y=0\text{and}x=0$

$y=4-\frac{1}{2}x,x=0,\text{and}y=0$

$y=2x^3,x=0,x=1,\text{and}y=0$

$y=3x^2,x=0,\text{and}y=3$

$y=\sqrt{4-x^2},y=0,\text{and}x=0$

$y=\frac{1}{\sqrt{x+1}},x=0,\text{and}x=3$

$x=\text{sec}\left(y\right)\text{and}y=\frac{\pi }{4},y=0\text{and}x=0$

$y=\frac{1}{x+1},x=0,\text{and}x=2$

$y=4-x,y=x,\text{and}x=0$

$y=x+2,y=x+6,x=0,\text{and}x=5$

$y=x^2\text{and}y=x+2$

$x^2=y^3\text{and}{x}^3=y^2$

$y=4-x^2\text{and}y=2-x$

[T] $y=\text{cos}x,y=e^{\text{−}x},x=0,\text{and}x=1.2927$

$y=\sqrt{x}\text{and}y=x^2$

$y=\text{sin}x\text{,}y=5\text{sin}x,x=0\text{and}x=\pi$

$y=\sqrt{1+x^2}\text{and}y=\sqrt{4-x^2}$

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

$y=x+2,y=2x-1,\text{and}x=0$

$y=\sqrt[3]{x}\text{and}y=x^3$

$x=e^{2y},x=y^2,y=0,\text{and}y=\text{ln}\left(2\right)$

$x=\sqrt{9-y^2},x=e^{\text{−}y},y=0,\text{and}y=3$

Yogurt containers can be shaped like frustums. Rotate the line $y=\frac{1}{m}x$ around the y -axis to find the volume between $y=a\text{and}y=b.$

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.

A better approximation of the volume of a football is given by the solid that comes from rotating $y=\text{sin}x$ 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 $y=\text{sin}x$ around the y -axis from $x=0$ to $x=\pi ?$

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.

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.

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.