A) Evaluate the line integral ?C xy^2 dy

a) Evaluate the line integral ?_C xy^2 dy - x^2 y dx where C is the circle x^2 + y^2 = a^2. Hint : Use Green's formula. b) Consider the vector field F(x, y) = (2xe^-y, 2y - x^2 e^-y). (i) Verify that F is a conservative vector field. (ii) Hence evaluate the line integral ?_C F(x, y) · dr, where C is any path from (1, 0) to (2, 1).

Transcript

00:01 Hello everyone, we have to evaluate the line integral.

00:04 So, which is equal to integral z minus x square y dx plus xy square dy is equal to integral c.

00:15 We can write it as p dx plus q dy form.

00:21 So, here p is equal to minus x square y and q is equal to x y square.

00:31 So, then dou q by dou x is equal to dou by dou x of xy square which is equal to y square.

00:43 Similarly, dou p by dou x is equal to minus x square.

00:48 So, now dou q by dou x minus dou p by dou y is equal to y square minus of minus x square which is equal to x square plus y square and now, c is the circle.

01:12 So, they given x square plus y square is equal to a square.

01:16 So, r of theta is equal to r cos theta comma r sin theta.

01:24 So, here theta is lies between 0 less than theta less than equal to 2 pi and r is lies between 0 to a.

01:35 So, and so d a is equal to r dr d theta.

01:41 So, now integral c xy square dy minus x square y dx is equal to double integral of dou q by dou x minus dou p by dou a of d a.

01:58 So, which is equal to double integral of x square plus y square d a.

02:04 So, which is equal to integral 0 to 2 pi integral 0 to a.

02:09 So, r power r r into dr d theta.

02:19 So, which is which is equal to integral 0 to 2 pi r power 4 by 4 0 to a at d theta.

02:29 So, it is equal to a power 4 by 4 into 2 pi.

02:33 Therefore, which is equal to pi a power 4 by 2 and now, we have to prove the second subdivision which is f of x comma y is equal to 2 x e power minus y comma 2 y minus x square e power minus y...

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