Review 5: Solids of Revolution - Now You Try I

1. What is the corresponding definite integral for the Riemann sum that we created in the example?

Answer:

▶ Show

$\lim_{n \to \infty} \sum_{i=1}^n \pi \left(x_i^* - (x_i^*)^2 \right)^2 \Delta x = \pi \int_0^1 (x-x^2)^2 \,dx.$ $\lim_{n \to \infty} \sum_{i=1}^n \pi \left(x_i^* - (x_i^*)^2 \right)^2 \Delta x = \pi \int_0^1 (x-x^2)^2 \,dx.$

Note that the bounds x=0,1 $x=0,1$ come from the intersection of the curve with the $x$ -axis.

2. What is the volume of the solid of revolution created by rotating the graph of y=x^2-x $y=x^2-x$ around the $x$ -axis?

Answer:

▶ Show

The volume should be given by:

$\pi \int_0^1 (x-x^2)^2\, dx = \pi \int_0^1 x^2-2x^3+x^4 \, dx = \frac{\pi}{30}.$ $\pi \int_0^1 (x-x^2)^2\, dx = \pi \int_0^1 x^2-2x^3+x^4 \, dx = \frac{\pi}{30}.$