(a) The proof that circle generated by rotating a point ( 0 , a , b ) about the z − a x i s is parameterized by ( a cos θ , a sin θ , b ) , ( 0 ≤ θ ≤ 2 π ) . (b) The proof that S is parameterized by G ( y , θ ) = ( y cos θ , y sin θ , g ( y ) ) , f o r ( c ≤ y ≤ d , 0 ≤ θ ≤ 2 π ) . (c) The proof of A r e a ( S ) = 2 π ∫ c d y 1 + [ g ' ( y ) ] 2 d y .

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Answer 1

Question

Chapter 17.4, Problem 42E

To determine

(a) The proof that circle generated by rotating a point $(0, a, b)$ about the $z - a x i s$ is parameterized by $(a \cos θ, a \sin θ, b), (0 \leq θ \leq 2 π)$ .

(b) The proof that $S$ is parameterized by $G (y, θ) = (y \cos θ, y \sin θ, g (y)), f o r (c \leq y \leq d, 0 \leq θ \leq 2 π)$ .

(c) The proof of $A r e a (S) = 2 π \int_{c}^{d} y \sqrt{1 + {[g' (y)]}^{2}} d y$ .

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Answer 2

Question

Chapter 17.4, Problem 42E

To determine

(a) The proof that circle generated by rotating a point $(0, a, b)$ about the $z - a x i s$ is parameterized by $(a \cos θ, a \sin θ, b), (0 \leq θ \leq 2 π)$ .

(b) The proof that $S$ is parameterized by $G (y, θ) = (y \cos θ, y \sin θ, g (y)), f o r (c \leq y \leq d, 0 \leq θ \leq 2 π)$ .

(c) The proof of $A r e a (S) = 2 π \int_{c}^{d} y \sqrt{1 + {[g' (y)]}^{2}} d y$ .

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Answer 3

Question

Chapter 17.4, Problem 42E

To determine

(a) The proof that circle generated by rotating a point $(0, a, b)$ about the $z - a x i s$ is parameterized by $(a \cos θ, a \sin θ, b), (0 \leq θ \leq 2 π)$ .

(b) The proof that $S$ is parameterized by $G (y, θ) = (y \cos θ, y \sin θ, g (y)), f o r (c \leq y \leq d, 0 \leq θ \leq 2 π)$ .

(c) The proof of $A r e a (S) = 2 π \int_{c}^{d} y \sqrt{1 + {[g' (y)]}^{2}} d y$ .

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Answer 4

Question

Chapter 17.4, Problem 42E

To determine

(a) The proof that circle generated by rotating a point $(0, a, b)$ about the $z - a x i s$ is parameterized by $(a \cos θ, a \sin θ, b), (0 \leq θ \leq 2 π)$ .

(b) The proof that $S$ is parameterized by $G (y, θ) = (y \cos θ, y \sin θ, g (y)), f o r (c \leq y \leq d, 0 \leq θ \leq 2 π)$ .

(c) The proof of $A r e a (S) = 2 π \int_{c}^{d} y \sqrt{1 + {[g' (y)]}^{2}} d y$ .

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Answer 5

Chapter 17.4, Problem 42E

To determine

(a) The proof that circle generated by rotating a point $(0, a, b)$ about the $z - a x i s$ is parameterized by $(a \cos θ, a \sin θ, b), (0 \leq θ \leq 2 π)$ .

(b) The proof that $S$ is parameterized by $G (y, θ) = (y \cos θ, y \sin θ, g (y)), f o r (c \leq y \leq d, 0 \leq θ \leq 2 π)$ .

(c) The proof of $A r e a (S) = 2 π \int_{c}^{d} y \sqrt{1 + {[g' (y)]}^{2}} d y$ .

Answer 6

Chapter 17.4, Problem 42E

To determine

(a) The proof that circle generated by rotating a point $(0, a, b)$ about the $z - a x i s$ is parameterized by $(a \cos θ, a \sin θ, b), (0 \leq θ \leq 2 π)$ .

(b) The proof that $S$ is parameterized by $G (y, θ) = (y \cos θ, y \sin θ, g (y)), f o r (c \leq y \leq d, 0 \leq θ \leq 2 π)$ .

(c) The proof of $A r e a (S) = 2 π \int_{c}^{d} y \sqrt{1 + {[g' (y)]}^{2}} d y$ .

Answer 7

Chapter 17.4, Problem 42E

To determine

(a) The proof that circle generated by rotating a point $(0, a, b)$ about the $z - a x i s$ is parameterized by $(a \cos θ, a \sin θ, b), (0 \leq θ \leq 2 π)$ .

(b) The proof that $S$ is parameterized by $G (y, θ) = (y \cos θ, y \sin θ, g (y)), f o r (c \leq y \leq d, 0 \leq θ \leq 2 π)$ .

(c) The proof of $A r e a (S) = 2 π \int_{c}^{d} y \sqrt{1 + {[g' (y)]}^{2}} d y$ .

Answer 8

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Answer 9

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Answer 10

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Answer 11

Ch. 17.1 - Prob. 1PQ Ch. 17.1 - Prob. 2PQ Ch. 17.1 - Prob. 3PQ Ch. 17.1 - Prob. 4PQ Ch. 17.1 - Prob. 1E Ch. 17.1 - Prob. 2E Ch. 17.1 - Prob. 3E Ch. 17.1 - Prob. 4E Ch. 17.1 - Prob. 5E Ch. 17.1 - Prob. 6E

Solution Summary: The author explains how the circle generated by rotating a point about the z a x i s is parameterized by cos, sin, and b.

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Solution Summary: The author explains how the circle generated by rotating a point about the z a x i s is parameterized by cos, sin, and b.

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(a) The proof that circle generated by rotating a point (0,a,b) about the z−axis is parameterized by (acosθ,asinθ,b),(0≤θ≤2π). (b) The proof that S is parameterized by G(y,θ)=(ycosθ,ysinθ,g(y)),for(c≤y≤d,0≤θ≤2π). (c) The proof of Area(S)=2π∫cdy1+[g'(y)]2dy.

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