6.3 Volumes of revolution: cylindrical shells  (Page 5/6)

[T] Over the curve of $y=3x,x=0,$ and $y=3$ rotated around the $y\text{-axis}.$

[T] Under the curve of $y=3x,x=0,\text{and}x=3$ rotated around the $y\text{-axis}.$

[T] Over the curve of $y=3x,x=0,\text{and}y=3$ rotated around the $x\text{-axis}.$

[T] Under the curve of $y=3x,x=0,\text{and}x=3$ rotated around the $x\text{-axis}.$

[T] Under the curve of $y=2x^3,x=0,\text{and}x=2$ rotated around the $y\text{-axis}.$

[T] Under the curve of $y=2x^3,x=0,\text{and}x=2$ rotated around the $x\text{-axis}.$

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

$y=5x^3,x=0,\text{and}x=1$

$y=\frac{1}{x},x=1,\text{and}x=100$

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

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

$y=\text{sin}{x}^2,x=0,\text{and}x=\sqrt{\pi }$

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

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

$y={\left(1+x^2\right)}^3,x=0,\text{and}x=1$

$y=5x^3-2x^4,x=0,\text{and}x=2$

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

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

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

$y=\text{ln}\left(x\right),x=1,\text{and}x=e$

$x=\frac{1}{1+y^2},y=1,\text{and}y=4$

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

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

$x=y^3-4y^2,x=\mathrm{-1},\text{and}x=2$

$x=y{e}^y\text{,}x=\mathrm{-1},\text{and}x=2$

$x=\text{cos}y{e}^y,x=0,\text{and}x=\pi$

$y=3-x,y=0,x=0,\text{and}x=2$ rotated around the $y\text{-axis}.$

$y=x^3,y=0,\text{and}y=8$ rotated around the $y\text{-axis}.$

$y=x^2,y=x,$ rotated around the $y\text{-axis}.$

$y=\sqrt{x},x=0,\text{and}x=1$ rotated around the line $x=2.$

$y=\frac{1}{4-x},x=1,\text{and}x=2$ rotated around the line $x=4.$

$y=\sqrt{x}\text{and}y=x^2$ rotated around the $y\text{-axis}.$

$y=\sqrt{x}\text{and}y=x^2$ rotated around the line $x=2.$

$x=y^3,y=\frac{1}{x},x=1,\text{and}y=2$ rotated around the $x\text{-axis}.$

$x=y^2\text{and}y=x$ rotated around the line $y=2.$

[T] Left of $x=\text{sin}\left(\pi y\right),$ right of $y=x,$ around the $y\text{-axis}.$

[T] $y=x^2$ and $y=4x$ rotated around the $y\text{-axis}.$

[T] $y=\text{cos}\left(\pi x\right),y=\text{sin}\left(\pi x\right),x=\frac{1}{4},\text{and}x=\frac{5}{4}$ rotated around the $y\text{-axis}.$

[T] $y=x^2-2x,x=2,\text{and}x=4$ rotated around the $y\text{-axis}.$

[T] $y=x^2-2x,x=2,\text{and}x=4$ rotated around the $x\text{-axis}.$

[T] $y=3x^3-2,y=x,\text{and}x=2$ rotated around the $x\text{-axis}.$

[T] $y=3x^3-2,y=x,\text{and}x=2$ rotated around the $y\text{-axis}.$

[T] $x=\text{sin}\left(\pi y^2\right)$ and $x=\sqrt{2}y$ rotated around the $x\text{-axis}.$

[T] $x=y^2,x=y^2-2y+1,\text{and}x=2$ rotated around the $y\text{-axis}.$

Use the method of shells to find the volume of a sphere of radius $r.$

Use the method of shells to find the volume of a cone with radius $r$ and height $h.$

Use the method of shells to find the volume of an ellipse $\left(x^2\text{/}{a}^2\right)+\left(y^2\text{/}{b}^2\right)=1$ rotated around the $x\text{-axis}.$

Use the method of shells to find the volume of a cylinder with radius $r$ and height $h.$

Use the method of shells to find the volume of the donut created when the circle $x^2+y^2=4$ is rotated around the line $x=4.$

Consider the region enclosed by the graphs of $y=f\left(x\right),y=1+f\left(x\right),x=0,y=0,$ and $x=a>0.$ What is the volume of the solid generated when this region is rotated around the $y\text{-axis}?$ Assume that the function is defined over the interval $[0,a].$

Consider the function $y=f\left(x\right),$ which decreases from $f\left(0\right)=b$ to $f\left(1\right)=0.$ Set up the integrals for determining the volume, using both the shell method and the disk method, of the solid generated when this region, with $x=0$ and $y=0,$ is rotated around the $y\text{-axis}.$ Prove that both methods approximate the same volume. Which method is easier to apply? ( Hint: Since $f\left(x\right)$ is one-to-one, there exists an inverse $f^{\mathrm{-1}}(y).)$