# What is the surface area produced by rotating #f(x)=e^(x^2), x in [-1,1]# around the x-axis?

As it turns out, the differential surface can be treated as the arc length for an infinitesimal distance along the chosen axis. So:

and:

First, let's evaluate the squared derivative:

This gives:

By signing up, you agree to our Terms of Service and Privacy Policy

To find the surface area produced by rotating ( f(x) = e^{x^2} ) around the x-axis over the interval ([-1,1]), we use the formula for the surface area of a solid of revolution:

[ S = \int_{a}^{b} 2\pi f(x) \sqrt{1 + \left( f'(x) \right)^2} , dx ]

where ( f'(x) ) is the derivative of ( f(x) ).

First, we find ( f'(x) ): [ f'(x) = 2xe^{x^2} ]

Next, we plug ( f(x) ) and ( f'(x) ) into the formula and integrate over the interval ([-1,1]): [ S = \int_{-1}^{1} 2\pi e^{x^2} \sqrt{1 + (2xe^{x^2})^2} , dx ]

[ = \int_{-1}^{1} 2\pi e^{x^2} \sqrt{1 + 4x^2e^{2x^2}} , dx ]

This integral may not have a closed-form solution and may need to be evaluated numerically.

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- How do you find the general solution to #dy/dx=xe^y#?
- 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 = sqrt(x#), #y = 0#, #y = 12 - x# rotated about the x axis?
- What is the surface area of the solid created by revolving #f(x)=x-1# for #x in [1,2]# around the x-axis?
- What is the arc length of #f(x)=10+x^(3/2)/2# on #x in [0,2]#?
- What is the average value of a function #f(x) = 2x sec2 x# on the interval #[0, pi/4]#?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7