use the shell method to find the volume of the solid generated by revolving the shaded region about the x…

use the shell method to find the volume of the solid generated by revolving the shaded region about the x - axis.\nset up the integral that gives the volume of the solid.\n\\(\\int_{0}^{\\square} \\square dy\\)

use the shell method to find the volume of the solid generated by revolving the shaded region about the x - axis.\nset up the integral that gives the volume of the solid.\n\\(\\int_{0}^{\\square} \\square dy\\)

Answer

Explanation:

Step1: Recall shell - method formula

The shell - method formula for volume when revolving about the x - axis is $V = 2\pi\int_{a}^{b}y\cdot h(y)dy$, where $y$ is the distance from the axis of revolution, and $h(y)$ is the height of the shell at $y$.

Step2: Determine the limits of integration

The region is bounded by $y = 0$ and $y=\sqrt{2}$. So, $a = 0$ and $b=\sqrt{2}$.

Step3: Find the height of the shell

The right - hand curve is $x = 4$ and the left - hand curve is $x = 2y^{2}$. The height of the shell $h(y)=4 - 2y^{2}$.

Step4: Set up the integral

Substitute $y$ and $h(y)$ into the shell - method formula: $V=2\pi\int_{0}^{\sqrt{2}}y(4 - 2y^{2})dy$.

Answer:

$2\pi\int_{0}^{\sqrt{2}}y(4 - 2y^{2})dy$