Compute the surface area of the torus in Exercise 45 using Pappus's Theorem. Data From Exercise 45

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Compute the surface area of the torus in Exercise 45 using Pappus's Theorem.

a b |L


Data From Exercise 45

Area of a Torus Let \(\mathcal{T}\) be the torus obtained by rotating the circle in the \(y z\)-plane of radius \(a\) centered at \((0, b, 0)\) about the \(z\)-axis (Figure 23). We assume that \(b>a>0\).

(a) Use Eq. (14) to show that
\[
\operatorname{area}(\mathcal{T})=4 \pi \int_{b-a}^{b+a} \frac{a y}{\sqrt{a^{2}-(b-y)^{2}}} d y
\]

af  x   + area(S) = 2n 1+g'(y) dy

(b) Show that \(\operatorname{area}(\mathcal{T})=4 \pi^{2} a b\).

x b-a b+a f b X

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Calculus

ISBN: 9781319055844

4th Edition

Authors: Jon Rogawski, Colin Adams, Robert Franzosa

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