Calculus
- (10 points) The equation of the red curve is y2 = x3. Find the volume generated when area bounded by (0,0), (4,8), (4,0) is revolved about the x-axis. (similar to 5.1 #7-12)
Sample Answer
To find the volume of the solid generated by revolving the area bounded by (0,0), (4,8), (4,0) about the x-axis, we can use the disk method.
Step 1: Set up the integral:
The disk method formula for finding the volume of a solid of revolution is:
V = π ∫[a,b] (R(x))^2 dx
where:
V
is the volume of the solidR(x)
is the radius of the disk at a given x-valuea
andb
are the limits of integration
In this case, R(x) = y = x^(3/2)
. The limits of integration are a = 0
and b = 4
.
Step 2: Evaluate the integral: