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Finding volume using Cylindrical Shells
- Thread starter rj686
- Start date
Hello, rj686!
I assume that all of these are to be done by "shells".
Shells Formula: \(\displaystyle \L\:V\;=\;2\pi\int^{\;\;\;b}_a(\text{radius})(\text{height}))\,dy\)
We are given: \(\displaystyle \;\begin{Bmatrix}\text{ Upper }y:\;y\,=\,2\sqrt{x} \\ \text{Lower }y:\;y\,=\,x\end{Bmatrix}\:\) and \(\displaystyle \:\begin{Bmatrix}\text{Right }x:\;x\,=\,y \\ \text{ Left }x:\;x\,=\,y^2/4\end{Bmatrix}\)
(A) About the x-axis: "horizontal" shells.
\(\displaystyle \L\;\;V\;=\;2\pi\int^{\;\;\;4}_0y\left(y\,-\,\frac{1}{4}y^2\right)\,dy\)
(B) About the y-axis: "vertical" shells.
\(\displaystyle \L\;\;V\;=\;2\pi\int^{\;\;\;4}_0 x\left(2\sqrt{x}\,- \,x\right)\,dx\)
(C) About \(\displaystyle x\,=\,4\): "vertical" shells.
\(\displaystyle \L\;\;V\;=\;2\pi\int^{\;\;\;4}_0(4 - x)(2\sqrt{x}\,-\,x)\,dx\)
(D) About \(\displaystyle y\,=\,4\): "horizontal" shells.
\(\displaystyle \L\;\;V\;=\;2\pi\int^{\;\;\;4}_0(4\,-\,y)\left(y\,-\,\frac{1}{4}y^2\right)\,dy\)
. . . someone check my work, please!
I assume that all of these are to be done by "shells".
Find the volume of the silid generated by revolving the region enclosed by
the parabola y^2=4x and the line y=x about:
A) the x-axis\(\displaystyle \;\;\;\)B) the y-axis\(\displaystyle \;\;\;\)C) the line x=4\(\displaystyle \;\;\;\)D) the line y=4
Code:
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--*--------------Shells Formula: \(\displaystyle \L\:V\;=\;2\pi\int^{\;\;\;b}_a(\text{radius})(\text{height}))\,dy\)
We are given: \(\displaystyle \;\begin{Bmatrix}\text{ Upper }y:\;y\,=\,2\sqrt{x} \\ \text{Lower }y:\;y\,=\,x\end{Bmatrix}\:\) and \(\displaystyle \:\begin{Bmatrix}\text{Right }x:\;x\,=\,y \\ \text{ Left }x:\;x\,=\,y^2/4\end{Bmatrix}\)
(A) About the x-axis: "horizontal" shells.
\(\displaystyle \L\;\;V\;=\;2\pi\int^{\;\;\;4}_0y\left(y\,-\,\frac{1}{4}y^2\right)\,dy\)
(B) About the y-axis: "vertical" shells.
\(\displaystyle \L\;\;V\;=\;2\pi\int^{\;\;\;4}_0 x\left(2\sqrt{x}\,- \,x\right)\,dx\)
(C) About \(\displaystyle x\,=\,4\): "vertical" shells.
\(\displaystyle \L\;\;V\;=\;2\pi\int^{\;\;\;4}_0(4 - x)(2\sqrt{x}\,-\,x)\,dx\)
(D) About \(\displaystyle y\,=\,4\): "horizontal" shells.
\(\displaystyle \L\;\;V\;=\;2\pi\int^{\;\;\;4}_0(4\,-\,y)\left(y\,-\,\frac{1}{4}y^2\right)\,dy\)
. . . someone check my work, please!
Here's the disk method then:
about x-axis:
\(\displaystyle \L\\{\pi}\int_{0}^{4}(4x-x^{2})dx\)
about y-axis:
\(\displaystyle \L\\{\pi}\int_{0}^{4}((\frac{y^{2}}{4})^{2}-y^{2})dy\)
about x=4:
\(\displaystyle \L\\{\pi}\int_{0}^{4}((\frac{y^{2}}{4}-4)^{2}-(y-4)^{2})dy\)
about y=4:
\(\displaystyle \L\\{\pi}\int_{0}^{4}((x-4)^{2}-(\sqrt{4x}-4)^{2})dx\)
Our respective methods jive, Soroban. That's either a good thing or real dumb luck. :lol:
about x-axis:
\(\displaystyle \L\\{\pi}\int_{0}^{4}(4x-x^{2})dx\)
about y-axis:
\(\displaystyle \L\\{\pi}\int_{0}^{4}((\frac{y^{2}}{4})^{2}-y^{2})dy\)
about x=4:
\(\displaystyle \L\\{\pi}\int_{0}^{4}((\frac{y^{2}}{4}-4)^{2}-(y-4)^{2})dy\)
about y=4:
\(\displaystyle \L\\{\pi}\int_{0}^{4}((x-4)^{2}-(\sqrt{4x}-4)^{2})dx\)
Our respective methods jive, Soroban. That's either a good thing or real dumb luck. :lol: