Evaluate R y sin(xy) dA, where R =[5,7][0,]. Solution 1 If we first integrate with respect to x, we get the following. R y sin(xy) dA =075 y sin(xy) dx dy =0cos(xy) x =7x =5dy =0cos(7y)+cos(5y)dy =sin(7y)7+sin(5y)50=0 Very nice! Solution 2 If we reverse the order of integration, we get the following. R y sin(xy) dA =750 y sin(xy) dy dx To evaluate the inner integral, we use integration by parts with u=ydv= sin(xy) dy du=dyv= cos(xy)x and so 0 y sin(xy) dy = y cos(xy)x y =y =0+1x 0 cos(xy)dy = cos(x)x +1x2 sin(xy) y =y =0= cos(x)x + sin(x)x2 Excellent job! If we now integrate the first term by parts with u =1x and dv = cos(x) dx, we get du =1x2 dx, v = sin(x), and cos(x)x dx = cos(x)2x sin(x)x2dx. Therefore cos(x)x + sin(x)x2dx = ycos(xy) Check your answer by taking the derivative. and so 750 y sin(xy) dy dx= sin(x)x275=0.