Studdy Solution
STEP 1
1. The region is bounded by the curves y=x2 and y=1.
2. The region is rotated about the line y=2.
STEP 2
1. Determine the points of intersection.
2. Set up the integral using the washer method.
3. Evaluate the integral to find the volume.
STEP 3
Determine the points of intersection of the curves y=x2 and y=1.
Set the equations equal to each other to find the intersection points:
x2=1
Solve for x:
x=±1
The points of intersection are x=−1 and x=1.
STEP 4
Set up the integral using the washer method. The outer radius R(x) is the distance from the line y=2 to y=x2, and the inner radius r(x) is the distance from y=2 to y=1.
R(x)=2−x2
r(x)=2−1=1
The volume V is given by:
V=π∫−11(R(x)2−r(x)2)dx
Substitute the expressions for R(x) and r(x):
V=π∫−11((2−x2)2−12)dx
STEP 5
Evaluate the integral:
First, expand the integrand:
(2−x2)2=4−4x2+x4
So the integral becomes:
V=π∫−11(4−4x2+x4−1)dx
=π∫−11(3−4x2+x4)dx
Now, integrate term by term:
V=π[3x−34x3+51x5]−11
Evaluate the definite integral:
V=π([3(1)−34(1)3+51(1)5]−[3(−1)−34(−1)3+51(−1)5])
=π([3−34+51]−[−3+34−51])
=π([1545−1520+153]−[−1545+1520−153])
=π(1528+1562)
=π(1590)
=6π
The volume of the solid is:
6π
