Using Green's theorem, find (cos(x^5) + y) dx + (e^y^4 -...

Transcript

00:01 In this question we are given the integral as integral x e to the power minus 2x d x plus x to the power 4 plus 2 x to y square d y so we have to evaluate this integral using the green theorem and the equation of the circles are x squared plus y square is equal to 9 and x squared plus y square is equal to 64 so the greens theorem states at integral on the curve c f of x comma y dx plus g of x comma y dx plus g of x comma y dy is equal to double integral over the region are daver g by daver x minus devere f divided by devour y d a this is what the green theorem states so if we will put the function here suitably integration with the region c x e to the power minus 2x d x plus x4 x to the power 4 plus 2 x square y square d y will be equals to double integral of devor divided by daver x of g which is x to the power 4 plus 2 x square y square minus daver divided by daver y of x e to the power minus 2x and here it is d a so on the right hand side we will get on differentiating it with respect to x we will get double integral 4x cube plus here we will get 4x y square and here it will be become 0 and it will be da so on using the polar co.

02:27 Coordinates we will here get it equals to integral from 0 to 2 pi and here it will be the integral from so as the circle are from the radius 3 to the radius 8 so integral will be from 3 to 8 x cube plus 4 x cubed plus 4 x y square d a so on using the polar coordinates we'll get integral from 0 to 2 pi from 3 to 8 in place of x we can write r cos theta so it will be 4 r cube cos qeta plus 4 r cus r sine theta whole square r d r d theta t theta so this is what we have got as the integral so it will be equals to integral from 0 to 2 pi integral from 3 to 8 and and on integrating, on solving them further we will get 4 r to the power 4.

04:01 Here it will be 4 r to the power 4 cos cube theta, just 4 r to the power 4, cost theta, sine 2 theta, dr, d theta.

04:17 So it will be equal to so.

04:21 So on integrating it we will get integral from 0 to 2 pi and here inside we will get it as 4 divided by 5 r to the power 5 cos cube theta plus 4 divided by 5 r to the power 5 cost theta sine square theta and here it will be t theta and the limits will be from 3 to 8 so here it will be equals to integration from 0 to 2 pi and here we'll get 3 to 5 to 5 25 multiplied by 4 divided by 5 cos cube theta plus 32525 multiplied by 4 divided by 5 cos theta sine square theta d theta so on integrating it we will get it as 3252525 multiplied by 4 divided by 5 and here inside the integral it will be on evaluating the integral we will get here, sine theta minus sine cube theta plus sine cube theta from 0 to 2 pi.

06:11 So on putting the limits we'll get its value as equals to 0...

Question

Using Green's theorem, find \(\oint_D (\cos(x^5) + y) \, dx + (e^{y^4} - x) \, dy\), where D is the region enclosed by the circle $x^2 + y^2 = 8$.

Question

Using Green's theorem, find \(\oint_D (\cos(x^5) + y) \, dx + (e^{y^4} - x) \, dy\), where D is the region enclosed by the circle $x^2 + y^2 = 8$.

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