Question

Let ????=−5y????+5x????. Use the tangential vector form of Green's Theorem to compute the circulation integral ∫????????⋅???????? where C is the positively oriented circle x2+y2=25

Answer 1

The value of the required integral is 50π  where C is the positively oriented circle x2+y2=25

We are given a vector F = -yi + xj. A vector is a quantity with magnitude and direction that is commonly represented by a directed line segment, with length representing magnitude and orientation in space representing direction. We need to use the tangential vector form of Green's Theorem to compute the circulation integral ∫CF⋅dr where C is the positively oriented circle x² + y² = 25

Let the value of the integral be represented by the variable "I".

I = ∫∫[(/x)(x) - (/y)(-y)]dA

I = ∫∫[(1 - (-1)]dA

I = ∫∫2dA

I = 2∫∫dA

I = 2×(Area of the circle)

I = 2*(πr²)

I = 2*(π*25)

I = 50π

Hence, the value of the required integral is 50π.

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