Review 5: Solids of Revolution - Now You Try II

Consider the following Riemann sum:

$\sum_{i=1}^n 2 \pi x_i^* \left(1- (x_i^*)^2 \right) \Delta x.$ $\sum_{i=1}^n 2 \pi x_i^* \left(1- (x_i^*)^2 \right) \Delta x.$

What is the associated definite integral to the Riemann sum above?

Answer:

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

The bounds for the integral come from the domain specified for the function.

What is the volume of revolution for example 2?

Answer:

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We compute

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