A question on my math homework that I can't seem to solve...
Rotate the region bounded by y=x^2-3x and the x-axis about the line x=4. Set up the integral to find the volume of the solid.
I'm pretty sure that the integral is in terms of (dy), and has bounds of 0-4. Using the slice method, the radius is (4-x), but I need the radius in terms of y. I tried solving for x to use substitution, but it didn't work.
What would this equation be, solved for x, and what would the integral be for finding the volume?
lets do it as thin walled cylinders rather than as circular slices.
each cylinder is at height y and at radius (4 - x) with wall thickness dx
The cylinders start at x = 0 and end at x = 3 (where y = 0, we are looking at a sliced bagel with a donut hole)
the circumference of each cylinder is 2 pi r = 2 pi (4-x)
so
dV = dx *2pi *(4-x) (x^2-3x)
integrate from x = 0 to x = 3
To solve this problem, you need to find the equation for x in terms of y so that you can express the radius in terms of y. Let's start by rearranging the equation y = x^2 - 3x:
y = x(x - 3)
Now, let's solve this equation for x:
x^2 - 3x - y = 0
Using the quadratic formula:
x = [-(-3) ± √((-3)^2 - 4(1)(-y))] / (2(1))
x = [3 ± √(9 + 4y)] / 2
Since we need the radius in terms of y, we can rewrite this as:
x = (3 + √(9 + 4y)) / 2
Now that we have the equation for x in terms of y, we can proceed to set up the integral to find the volume.
The volume of the solid can be found using the disk method:
V = ∫[a, b] π(radius)^2 dy
In this case, the integral bounds are 0 to 4, as you correctly stated.
The radius is given by (4 - x), so substituting x = (3 + √(9 + 4y)) / 2:
V = ∫[0, 4] π((4 - [(3 + √(9 + 4y)) / 2]))^2 dy
Now, square the expression inside the parentheses and simplify:
V = ∫[0, 4] π(16 - 14(3 + √(9 + 4y)) + (3 + √(9 + 4y))^2) dy
V = ∫[0, 4] π(16 - 42 - 14√(9 + 4y) + 9 + 6√(9 + 4y) + 9 + 2(3 + √(9 + 4y))^2) dy
V = ∫[0, 4] π(34 + 4(3 + √(9 + 4y)) + (3 + √(9 + 4y))^2) dy
Now, you can evaluate this integral to find the volume of the solid.