How do you use the method of cylindrical shells to find the volume generated by rotating the region bounded by #y=e^(−x^2)#, y=0, x=0, and x=1 about the y axis?

Question

Evelyn Agard · Accepted Answer

Draw a sketch, then integrate using the shell method with respect to x.

Here is sketch of the problem:

Using the formula for the Shell Method:

#2piint_0^1(x)(e^(-x^2)-0)dx#

You can use substitution {#u=x^2#} to solve.

#=pi(1-e^-1)#

Answer #~~1.986#

Hope that helps

Aurora Alvarado · Answer

undefined To use the method of cylindrical shells to find the volume generated by rotating the region bounded by $ y = e^{-x^2} $, $ y = 0 $, $ x = 0 $, and $ x = 1 $ about the y-axis, follow these steps:

1. Determine the height of the cylindrical shells: The height of each cylindrical shell will be the difference between the y-value of the upper function ($ y = e^{-x^2} $) and the y-value of the lower function ($ y = 0 $). Therefore, the height will be $ e^{-x^2} - 0 = e^{-x^2} $.

2. Determine the radius of each cylindrical shell: The radius of each cylindrical shell will be the distance from the axis of rotation (y-axis) to the x-value of each shell, which is $ x $.

3. Set up the integral: The formula for the volume using cylindrical shells is $ V = \int 2\pi rh \, dx $, where $ r $ is the radius, $ h $ is the height, and $ dx $ denotes integration with respect to $ x $. So, the integral becomes:
$$ V = \int_{0}^{1} 2\pi x \cdot e^{-x^2} \, dx $$

4. Integrate: Use techniques of integration to evaluate the integral from 0 to 1.

5. Calculate: Once you have found the integral, evaluate it to find the volume.

By following these steps, you can use the method of cylindrical shells to find the volume generated by rotating the given region about the y-axis.