   Chapter 10.4, Problem 56E

Chapter
Section
Textbook Problem

(a) Find a formula for the area of the surface generated by rotating the polar curve r = f(θ), a ≤ θ ≤ b (where f′ is continuous and 0 ≤ a < b ≤ π), about the line θ = π/2. (b) Find the surface area generated by rotating the lemniscate r2 = cos 2θ about the line θ = π/2.

(a)

To determine

To find: The formula for the area of the surface generated by rotating the polar curve r=f(θ) with limit tends to be aθb (where f' is continuous and 0a<bπ ) about the line θ=π2 .

Explanation

Calculation:

When r=f(θ) is rotated by an angle α , then the value of r becomes as below.

r=f(θα)

When r=f(θ) is rotated by an angle π2 , then the value of r becomes as below.

r=f(θπ2)

Use the surface area formula obtained after rotating r=f(θ) about the polar line.

S(θ)=ab2πrsinθ(r2+(drdθ)2)dθ (1)

Replace θ with (θ+π2) in the equation (1)

(b)

To determine

To find: The surface area generated by rotating the lemniscate r2=cos2θ about the line θ=π2 .

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