Use Green's Theorem to evaluate ∫C F·dr. x?
Use Green's Theorem to evaluate ∫C F·dr. (Check the orientation of the curve before applying the theorem.)
F(x, y) = ‹e2x + x^(2)y, e^(2y) - xy^(2)›
C is the circle x2 + y2 = 16 oriented clockwise.
More Questions From This User See All
Answers & Comments
Verified answer
The component of the curl of F in the z-direction is
∂/∂x [e^(2y) - xy^(2)] - ∂/∂y [e2x + x^(2)y] = -y² - x².
The orientation is backward, so we integrate the negative of this---i.e. x² + y²---over the disk of radius 4. This is better done in polar coordinates. The disk is
0 ≤ Θ ≤ 2π, 0 ≤ r ≤ 4
and the integrand is r² with differential area r dr dΘ.
∫C F·dr =
2π..4
∫ ....∫ r^3 dr dΘ = 128π.
0 ..0
by using green's Theorem, the line severe would additionally be expressed because of fact the subsequent 2 variable vital: ?(2xy-2)dxdy As 2xy is the partial by using-made from x^2y with savour to x and 2 is the partial derivative of 2y with savour to y. ?(2xy)dxdy-?(2)dxdy because of symmetry of the sphere, the 1st term equals 0, and the 2d term comes out to be -2(component of the circle). for this reason, the line mandatory equals -eight?.