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We arrived at Formula 6.3.2, $ V = \displaystyle \int_a^b 2 \pi x f(x) dx $, by using cylindrical shells, but now we can use integration by parts to prove it using the slicing method of Section 6.2, at least for the case where $ f $ is one-to-one and therefore has an inverse function $ g $. Use the figure to show that$$ V = \pi b^2 d - \pi a^2 c - \int_c^d \pi [g(y)]^2 dy $$Make the substitution $ y = f(x) $ and then use integration by parts on the resulting integral to prove that$$ V = \int_a^b 2 \pi x f(x) dx $$

$V=$ $\int_{a}^{b} 2 \pi x f(x) d x$

Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 1

Integration by Parts

Integration Techniques

Missouri State University

Harvey Mudd College

University of Michigan - Ann Arbor

Boston College

Lectures

01:11

In mathematics, integratio…

06:55

In grammar, determiners ar…

01:37

Cylindrical shells In Sect…

16:38

a. Let $B$ be a cylindrica…

10:38

08:21

(a) Use integration by par…

08:59

Convert the integral$$…

05:59

Evaluate the integral by c…

03:55

Evaluate the following int…

03:28

Convert the integral $\int…

03:14

01:06

Use cylindrical or spheric…

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