Calculus on Demand at Dartmouth College Lecture 29 | Index Lecture 30

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In this lecture we learn that integrals can be used to compute volumes as well as areas. We apply the Method of Accumulations to develop a formula for the volume of a solid generated by revolving a plane region about the x-axis.

Quick Question

Rotate the following semicircular region about the x-axis. What figure results and what is its volume?

Textbook

Volumes of Solids of Revolution

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Quiz

Volumes of Solids of Revolution Quiz

Examples

• Find the volume of the solid generated by rotating the region R bounded by the y axis, the line y = a, and the curve y = √x around the x axis.
• Find the volume of the solid generated by rotating the region bounded by y = x, y = 3 – x, and x = 4 around the line x = 5.
• Find the volume of the torus of radius a with inside radius b.

Videos

• Find the volume of the cone created by rotating y = 3x around the x-axis between x = 0 and x = 1
• Find the volume of the solid created by rotating y = cos x around the x-axis between x = 0 and x = π/2

Lecture 29 | Index