Evaluate R y sin ( xy ) dA , where R = [ 5 , 7 ] [ 0 , ] . Solution 1 If
Evaluate R y sinxy dA where R Solution If we first integrate with respect to x we get the following. R y sinxy dA y sinxy dx dy cosxy x x dy cosycosydy sinysiny Very nice! Solution If we reverse the order of integration, we get the following. R y sinxy dA y sinxy dy dx To evaluate the inner integral, we use integration by parts with uydv sinxy dy dudyv cosxyx and so y sinxy dy y cosxyx y y x cosxydy cosxx x sinxy y y cosxx sinxx Excellent job! If we now integrate the first term by parts with u x and dv cosx dx we get du x dx v sinx and cosxx dx cosxx sinxxdx Therefore cosxx sinxxdx ycosxy Check your answer by taking the derivative. and so y sinxy dy dx sinxx
