Explanation Given Info: C R be a circle of radius R centred at the origin. Formula Used: Green’s theorem can be stated as follows. If M and N are functions of ( x , y ) defined on a region D bounded by closed curve C and have partial derivatives then, ∮ C M d x + N d y = ∬ D ( ∂ N ∂ x − ∂ M ∂ y ) d x d y Calculation: According to Green’s theorem, ∮ C R y 3 d x + x d y = ∬ ( 1 − 3 y 2 ) d x d y In polar co-ordinates, x = r cos θ , y = r sin θ and d x d y = r d r d θ Therefore, ∮ C R y 3 d x + x d y = ∫ 0 2 π ∫ 0 R ( 1 − 3 r 2 sin 2 θ ) r d r d θ = ∫ 0 2 π ∫ 0 R ( r − 3 r 3 sin 2 θ ) d r d θ = ∫ 0 2 π [ r 2 2 − 3 r 4 4 sin 2 θ ] 0 R d θ = ∫ 0 2 π [ R 2 2 − 3 R 4 4 sin 2 θ ] d θ = ∫ 0 2 π [ R

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ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 18.1, Problem 35E

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The value of $R$ which maximizes the integral $\oint_{C_{R}} y^{3} d x + x d y$ .

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Chapter 18.1, Problem 34E

Chapter 18.1, Problem 36E

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The value of R which maximizes the integral ∮ C R y 3 d x + x d y .

Solution Summary: The author explains how the value of R which maximizes the integral is R = 2 3 and the formula used to calculate it.

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The value of $R$ which maximizes the integral $\oint_{C_{R}} y^{3} d x + x d y$ .

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Chapter 18 Solutions

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Solution Summary: The author explains how the value of R which maximizes the integral is R = 2 3 and the formula used to calculate it.

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The value of $R$ which maximizes the integral $\oint_{C_{R}} y^{3} d x + x d y$ .