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Compute the area of the region bounded by 1= 2r and J=8.Area , A = f xdy-y dxFor the following vector field, compute (a) the circulation on and (b) the outward flu...

Question

Compute the area of the region bounded by 1= 2r and J=8.Area , A = f xdy-y dxFor the following vector field, compute (a) the circulation on and (b) the outward flux across the boundary of the given region. Assume the boundary curves have counter-clockwise orientation. F-v(2 W+>1, where R is the half annulus {(r,0)13 <r<5,0 <0 < t}Use " Green's Theorem t0 evaluate the line integralTxy)dx + [Sx' + sin (y)] dy whereis the positively-oriented boundary of the region bou

Compute the area of the region bounded by 1= 2r and J=8. Area , A = f xdy-y dx For the following vector field, compute (a) the circulation on and (b) the outward flux across the boundary of the given region. Assume the boundary curves have counter-clockwise orientation. F-v(2 W+>1, where R is the half annulus {(r,0)13 <r<5,0 <0 < t} Use " Green's Theorem t0 evaluate the line integral Txy)dx + [Sx' + sin (y)] dy where is the positively-oriented boundary of the region bounded by the two cireles of radii of 2 and _ centered at the origin and lying in the first quadrant 10. Use parametric description of the sphere to find the surface area of the sphere of radius



Answers

General regions For the following vector fields, compute (a) the circulation on and (b) the outward flux across the boundary of the given region. Assume boundary curves are oriented counterclockwise. $\mathbf{F}=\langle 2 x+y, x-4 y\rangle ; R$ is the quarter-annulus $\{(r, \theta): 1 \leq r \leq 4,0 \leq \theta \leq \pi / 2\}$.

We have. If fixing Roy equals X G Dixon Roy equals work. This information is oversea if we are equals integral for our the partial derivative G with respect to X minus the partial derivative. If with this pick Troy de that equals the integral for wolf one plus one de that equals the integration for off to doings. The square boy minus one is credible. So the sequence to 16 point where the flex is There you go oversea if be not in d s equals in particular We are off the puncher Deal with you if with respect to X plus the partial derivative g with respect oy knee that equals the integral Over our well minus one plus one d The sequence to integral over our zero a d a which equals toe zero. Thank you.

We have. If Nixon y equals six and G Well, Nixon Roy equals what the circulation is, Diggle or C, If you're equals integral over our wolf, What should everything g with respect to X plus the pressure to deliver to If this picture Roy de that equals the integral off zero minus you, me, which equals zero the flexes, Tyga or C there look in B s equals Diggle are what should revert you if with respect to weeks. Thus the partial derivative key the specter boy d that equals integral who are one plus one detainee t a quest will digger what are well, toe de that equals two times 1/2 after blood by two square boy minus 1/2 multiplied by one square boy the equals two you boy. Thank you.

A problem number 14. To solve this exercise, we reduce the extension off the Green Theater, taking into account this year on the line. Integral can be written ad integration. If not, they are is integration off? See one at the AR minus integration off c. Two f d r. We're C one is an external boundary and see to it the internal boundary He's given the Victorian feed. So off x Why is equal to eat for negative X three wide off I yes x off Jane. We use the full informatization for the curve. C one X is equal to four course or in tea. Why she gets full saw him a team tear from 0 to 2 points. C two x is equal to one plus to consign See while she ableto sigh Inti from you 30 to find and then the integration off dot d r is c one f dot br When I see two f dot we are which is equal to integration off. See one off important negative 14 plus three wide off the D. X off team plus X off team Uh, the why well team minus integration off c two people negative. Exact E plus three wide off team the X off team. Yeah, X off T D y off T, which is equal to negative 24 burn.

All right. 35 Could be the Howard Flux for the following vector field Exxon. Why made, of course, on why the sign Why and s is the boundary The region bounded by X equals one y equals zero like ALS pi over two Z equals zero and Z equals X. All right, so finding the Grady int first we get great and zygote A three sign of why that's just taking the partial of X parts of wine partials you. So now we can set up our inner grow for the flux. Could be integral from zero to pi over two and the girl from 0 to 1 The girl from zero to x three Sign Why easy to you I t x Okay, this equals, you know, girl from zero to pi over two and a girl from 0 to 1 a three sign why Z Evaluated from zero to x dx dy why that's equal to in a grove power to in a girlfriend 01 three X sign Why d X d? Why further simplifying. Get integral from power for two of three times X squared over two 01 a sign Why do you Why cools 3/2 tones and a good co sign. Why that I waited from zero to pi over two, which is you go to three or two.


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