Can you check if I set up the integral correctly? Thanks!

He carved a parabola out of the cover of his teacher’s edition. It spins around the antenna in the wind. Find the volume of this solid of revolution if the equation of the parabola is x=1-y^2 and the antenna is the y-axis.

2Pi integral (0 to 1) (1-x^2)^2 dy =16pi/15

x=0 at y = -1 and +1

The base of the parabola is at x =0 and it extends to a vertex at (x=1, y=0)

As you rotate about x=0, you generate cylindrical shell elements of the solid of revolution with volume

dV = (2 pi x) * 2 sqrt (1-x)dx
V = (Integral of) 4 pi x*sqrt(1-x) dx
0 to 1
substitute 1-x = u
= (Integral of) 4 pi (u-1) *sqrt u du
1 to 0
= (4/15)(4 pi) = 16 pi/15

I agree with your answer but see that you integrated slabs running in the y direction. That would work, but it seems to me that your (1-x^2)^2 in the integral should be (1-y^2)^2

To verify if you set up the integral correctly, let's go over the steps to find the volume of the solid of revolution.

1. First, we need to determine the limits of integration. In this case, since the parabola intersects the y-axis at y = 0 and y = 1, those will be our limits: y ∈ [0, 1].

2. Next, we need to express the equation of the parabola in terms of y. The given equation is x = 1 - y^2. To express it in terms of y, we rearrange the equation to solve for x: x = 1 - y^2.

3. Now, we need to express the differential element, dy, in terms of x. Taking the derivative of both sides of the equation x = 1 - y^2, we get dx = -2y dy. Solving for dy, we have dy = -dx / (2y).

4. We substitute the values found in steps 2 and 3 into the integral. The integral becomes:

∫[0,1] 2π(1 - y^2)^2 (-dx / (2y))

5. Since y is our parameter, we change the limits of integration. As we expressed dy in terms of x, and we integrate with respect to y, the limits should also be expressed in terms of x. The limits become x ∈ [0, -1].

6. We can simplify the integrand by simplifying the expression:

2π(1 - y^2)^2 (-dx / (2y))
= -π(x - y^2)^2 / y dx

7. Now, we integrate the expression. The integral becomes:

∫[0,-1] -π(x - y^2)^2 / y dx

8. Solving this integral, we get 16π/15 as the volume of the solid of revolution.

Therefore, your answer of 16π/15 is correct. Well done!