How do you use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region #y=4-x^2# and #y=0# rotated about the y-axis?

Question

Christopher Agresta · Accepted Answer

See the explanation, below.

Here is the region with a thin slice taken parallel to the axis of rotation. (To set up for cylindrical shells.)

The volume of a representative shell is #2pirh*"thickness"#

In this case, we have radius #r = x#, height #h = 4-x^2# and #"thickness" = dx#. #x# varies from #0# to #2#, so the volume of the solid is:

#int_0^2 pi x(4-x^2)dx = piint_0^2 (4x-x^3) dx = 4pi#

Charles Arocha · Answer

undefined To use the shell method to set up and evaluate the integral for the volume of the solid generated by revolving the plane region $y = 4 - x^2$ and $y = 0$ rotated about the y-axis, you need to follow these steps:

1. Determine the limits of integration by finding the x-values where the curves intersect. In this case, solve $4 - x^2 = 0$ to find the bounds.

2. Set up the integral using the shell method formula: $V = 2\pi \int_{a}^{b} x \cdot h(x) \, dx$, where $h(x)$ represents the height of the shell.

3. Express $h(x)$ in terms of $x$ using the difference between the two curves: $h(x) = 4 - x^2 - 0 = 4 - x^2$.

4. Integrate from the lower limit to the upper limit obtained in step 1 using the formula from step 2.

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

So, the integral to find the volume using the shell method is:

$$ V = 2\pi \int_{-2}^{2} x \cdot (4 - x^2) \, dx $$

Now, integrate this expression to find the volume.