Question

Select all of the following integral(s) that represent the volume of the solid obtained by rotating the region bounded by x = 2y², y = 1, x = 0 about the y-axis.

A) ∫¹₀ 2πx(1 -√x/2)dx
B) ∫¹₀ 2πy(2y²)dy
C) ∫¹₀ π(1 -√x/2)²dx
D) ∫¹₀ 2πx(1 -√x/2)²dx
E) ∫¹₀ 2π(2y²)dy
F) ∫¹₀ π(2y²)²dy
G) ∫²₀ π(2y²)²dy

Answer 1

The integral that represents the volume of the solid obtained by rotating the region bounded by x = 2y², y = 1, and x = 0 about the y-axis is option D) ∫¹₀ 2πx(1 -√x/2)²dx.

Step-by-step explanation:

To find the volume of the solid obtained by rotating the region bounded by x = 2y², y = 1, and x = 0 about the y-axis, we can use the method of cylindrical shells.

The integral that represents the volume is given by option D) ∫¹₀ 2πx(1 -√x/2)²dx.

Let's break down the steps to solve this:

1. Express y in terms of x: y = √(x/2) (from x = 2y²)
2. Determine the limits of integration: x ranges from 0 to 1 (from x = 0 and x = 1)
3. Write the integral: ∫¹₀ 2πx(1 -√(x/2))²dx
4. Simplify the integrand: 2πx(1 -√(x/2))² = 2πx(1 - (x/2)) = 2πx(1 - x/2) = 2πx - πx²/2
5. Evaluate the integral: integrate 2πx - πx²/2 with respect to x from 0 to 1

The final answer is the result of evaluating the integral, which should be a numerical value.