The region bounded by y=x^2 and y=4 is rotated about the line y=-1. Find the volume.
To find the volume of the solid generated by rotating the region bounded by y=x^2 and y=4 about the line y=-1, we can use the method of cylindrical shells.
Let's break down the process step by step:
1. Determine the limits of integration:
To find the volume, we need to integrate with respect to x. First, let's find the limits of integration for x.
The curves y=x^2 and y=4 intersect at x=-2 and x=2. So, our limits of integration will be -2 to 2.
2. Set up the integral:
The formula for the volume using cylindrical shells is V = ∫2πxf(x)dx, where f(x) represents the distance between the line of rotation and the curve at each value of x.
In this case, the distance between the line y=-1 and the curve y=x^2 is given by:
f(x) = (x^2 - (-1)) = x^2 + 1.
Therefore, the integral for the volume becomes:
V = ∫[-2,2] 2πx(x^2 + 1)dx
3. Calculate the integral:
Evaluating the integral will give us the volume:
V = 2π ∫[-2,2] (2x^3 + x)dx
Solving this integral will give us the volume of the solid.
4. Evaluate the integral:
To evaluate the integral, we integrate each term separately:
V = 2π [ ∫(2x^3)dx + ∫(x)dx ]
V = 2π [ (1/2)x^4 + (1/2)x^2 ] evaluated from -2 to 2
Plugging the limits into the expression:
V = 2π [ ((1/2)(2)^4 + (1/2)(2)^2) - ((1/2)(-2)^4 + (1/2)(-2)^2) ]
Simplifying further:
V = 2π [ (1/2)(16+4) - (1/2)(16+4) ]
V = 2π [ (10) - (10) ]
V = 0
Therefore, the volume of the solid generated by rotating the region bounded by y=x^2 and y=4 about the line y=-1 is 0 units cubed.