Cylindrical shells trouble integrating

[ACLerok]

i'm told to use the method of cylindrical shells to find the volume gnerated by rotating the region bounded these curves about the y axis:
y=e^(-x^2)
y=0
x=0
x=1
Couple questions. What does the graph of y=e^(-x^2) look like? Also, i know the integral is =integral(a,b) 2(pi)x*e^(-x^2) how do i evaulate the e^(-x^2) part?

 

[HallsofIvy]

The graph of e^{-x^2} is the famous "Bell shaped curve"!

As far as the integral of x e^{-x^2} is concerned, try the substitution u= -x^2.

 

[M98Ranger]

The graph of y=e^(-x^2) is a bell-shaped curve that decreases rapidly as x increases. It is also known as a Gaussian curve or a normal distribution curve.

To evaluate the integral of e^(-x^2), you can use the substitution method. Let u=-x^2, then du=-2x dx. Substitute these into the integral to get:
integral(a,b) 2(pi)x*e^(-x^2) dx = integral(a,b) 2(pi) ue^u du
This can now be evaluated using the power rule for integrals. Once you have the antiderivative, you can substitute back in for u and evaluate the integral from a to b. Alternatively, you can use a calculator or a software program to evaluate the integral numerically.