How do you use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by #y = 2 - (x^2)# and #y = x^2# revolved about the x=1?

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Isabella Ackerman · Accepted Answer

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Isabella Ackerman · Answer

undefined To use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by $y = 2 - x^2$ and $y = x^2$ revolved about $x = 1$, follow these steps:

1. Determine the limits of integration by finding the points of intersection of the two curves:
   $2 - x^2 = x^2$
   Solve for $x$ to find the intersection points.

2. Set up the integral using the formula for cylindrical shells:
   $$V = 2\pi \int_{a}^{b} r(x) \cdot h(x) \, dx$$
   where $r(x)$ is the distance from the axis of rotation (in this case, $x = 1$) to the shell, and $h(x)$ is the height of the shell.

3. Calculate $r(x)$ and $h(x)$ in terms of $x$:
   - $r(x)$ is the distance from $x$ to the axis of rotation, which is $|x - 1|$.
   - $h(x)$ is the difference in $y$ values of the two curves, which is $(2 - x^2) - (x^2)$.

4. Integrate the expression obtained in step 2 from the lower limit of integration $a$ to the upper limit of integration $b$.

5. Evaluate the integral to find the volume of the solid.

That's how you use the method of cylindrical shells to find the volume of the solid obtained by rotating the given region about the given axis.