The solid obtained by rotating the region bounded by y=x^7, x=1, and y=−1 around the axis y=−1. Write a Riemann sum approximating the solid (use Dx for Δx ): volume ≈Σ

Question

I already found the sum to the integrals which is 32pi/5 however, I keep getting the Riemann sum volume wrong. Can somebody help me with this question.

*Then convert your sum to an integral and find the volume. volume = 32pi/5

Answer 1

For the Riemann Sum if you divide the interval from -1 to 1 into n equal sub-intervals, the width of each sub-interval will be 2/n,

If you use right hand end endpoints, x₁ = -1 + 1(2/n), that is, the value for x₁ = -1 + i sub- intervals.

In general, x_i = -1 + i(2/n) or -1 + 2i/n.

Essentially we are using the disk method to find the volume of a typical disk which is found by πR²H.

The radius of a typical disk is x⁷ - (-1), that is distance from the point on the curve to the axis of revolution.

Putting it all together:V = π ∑ⁿ_i=1 (2/n) ((2i/n -1)⁷+1)². If we find the limit as n-> infinity we will have the volume, but doing so would not be very much fun.

The 2/n represents the "height" of a typical disk, The 2nd parenthesis is the radius squared, where the value for x_i has been plugged into f(x).

Visit this Desmos link to compare the values of the Riemann sum as n gets big and the definite integral.

desmos.com/calculator/stgwbc9fmi