Math  /  Calculus

QuestionFind the volume VV of the solid obtained by rotating the region bounded by the given curves about the specified line., y=ex,y=0,x=2,x=2;y=e^{x}, \quad y=0, \quad x=-2, x=2 ; about the xx-axis V=V=

Studdy Solution

STEP 1

1. The region bounded by the curves y=ex y = e^x , y=0 y = 0 , x=2 x = -2 , and x=2 x = 2 is being rotated about the x x -axis.
2. The method of disks or washers is appropriate for finding the volume of the solid of revolution.

STEP 2

1. Set up the integral for the volume of the solid using the disk method.
2. Evaluate the integral to find the volume.

STEP 3

The volume V V of the solid obtained by rotating the region about the x x -axis can be found using the disk method. The formula for the volume is:
V=πab[f(x)]2dx V = \pi \int_{a}^{b} [f(x)]^2 \, dx
where f(x)=ex f(x) = e^x , a=2 a = -2 , and b=2 b = 2 .

STEP 4

Substitute the given function and limits into the formula:
V=π22(ex)2dx V = \pi \int_{-2}^{2} (e^x)^2 \, dx
Simplify the integrand:
V=π22e2xdx V = \pi \int_{-2}^{2} e^{2x} \, dx

STEP 5

Evaluate the integral:
The antiderivative of e2x e^{2x} is 12e2x \frac{1}{2} e^{2x} .
V=π[12e2x]22 V = \pi \left[ \frac{1}{2} e^{2x} \right]_{-2}^{2}

STEP 6

Substitute the limits of integration:
V=π(12e412e4) V = \pi \left( \frac{1}{2} e^{4} - \frac{1}{2} e^{-4} \right)
V=π2(e4e4) V = \frac{\pi}{2} \left( e^{4} - e^{-4} \right)
The volume of the solid is:
V=π2(e4e4) V = \frac{\pi}{2} \left( e^{4} - e^{-4} \right)

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