Verified answer

∮c (x^(2) y dx + 0 dy ) , Use Green Theo ,

=INT_A(0-x^2) dA = -INT _A x^2 dA

Polar coord

x=rcosT

dA=rdrdT

=-INT r^2cos^2T (rdrdT)

=-r^4/4 INT cos^2T dT

0<r<1

=-(1/4) INT cos^2T dT

=(-1/4) INT (1+cos2T)/2 dT

=(-1/8) (T+(1/2)sin2T)

0<T<2pi

=(-1/8) ( 2pi )

= -pi/4

So it looks like you want to compute the path integral of this function over the path of the unit circle if I understand this correctly. First I would convert everything to polar coordinates. In polar coordinates, r = sqrt(x^2+y^2) and theta = arctan(y/x). Rewrite your variables x and y in terms of these variables, then take your integral from theta = 0 to pi/2. The expression inside the integral will have two variables, r and theta. r will just = 1 because its the unit circle so your expression will just be in terms of theta. You also need to change dx to d(theta). Get an expression relating x to theta and r. r always = 1 so that will not be a problem. Differentiate your x(theta) expression in terms of theta to get dx/d(theta). Now you know the relationship between dx and d(theta). Convert all the x's inside your integral expression iinto r and theta terms. r = 1 as before and you're left only with theta terms to be integrated. Good luck.

Not going to try and solve it because I've forgotten how to calculate the Jacobian lol, but if I were given this question then I would try a change to polar co-ordinates so that your unit circle is just r = 1 (meaning that your limits of integration would be r between 0 and 1, and theta between 0 and 2pi), then making a change of variables from (x,y) to (r,theta), remembering to multiply by the Jacobian of the transformation. That should then give a pretty simple integral.

First you opt to remodel the curve to parametric type: x^2+y^2=9 => x=3cost, y=3sint, dx=3sint, dy=-3cost Now you've the ?y^3dy = ?(3sint)^3*3sint dt This imperative evaluates to three^4*a million/32*[12x - 8sin2x + sin4x] and think ofyou've got were given -?x^3dx = ?(3cost)^3*-3cost dt This evaluates to -3^4*a million/32*[12x + 8sin2x + sin4x] So your effect is -3^4*a million/32*[16sin2x + 2sin4x] desire that facilitates!