Question

Find the volume of the solid generated in the following situation: The region R in the first quadrant bounded by the graphs of y=x and y=x² is revolved about the line x.

Answer 1

To find the volume of the solid generated by revolving the region R about the line x, use the method of cylindrical shells. Integrate the expression 2πx times the difference between y = x and y = x² over the interval from x = 0 to x = 1. The volume of the solid is π/6 cubic units.

Step-by-step explanation:

Volume of the Solid Generated

To find the volume of the solid generated by revolving the region R about the line x, we can use the method of cylindrical shells. The volume can be calculated by integrating the circumference of a shell multiplied by its height over the interval where the two curves intersect.

1. First, we need to find the points of intersection between the two curves y = x and y = x². Setting the two equations equal to each other, we get x = x². Solving this equation, we find two intersection points: (0,0) and (1,1).
2. We have to rotate the region R about the line x, so the radius of the cylindrical shells will be x. The circumference of a shell is given by 2πx, and the height of each shell is the difference between the two curves at that x-value.
3. We need to integrate the expression 2πx times the difference between y = x and y = x² over the interval from x = 0 to x = 1: ∫01 2πx(x - x²) dx.
4. Simplifying the integral, we get ∫01 2π(x² - x³) dx.
5. Evaluating the integral, we get [2π(x³/3 - x⁴/4)]01 = 2π(1/3 - 1/4) = 2π(4/12 - 3/12) = 2π/12 = π/6.

Therefore, the volume of the solid generated by revolving region R about the line x is π/6 cubic units.