Consider the integral "S" f(x, y) dydz. Sketch the region of integration and change the order of integration. 10z •92(y) [ f(x, y)dxdy 91 (3) a = 91(y) b 92(y)

Consider the case when S[y] = [" √1+32 dx y(0) = 8, 0 y where y(v) > 0, 5 > 0 and the right-hand end point (v, y(v)) lies on the line ay +ẞx+y= 0, where a, ß, y are constants with ẞ +0. Show that the first-integral may be written as y√1+y = C for some constant c > 0, and that the solutions of the first-integral are circles centred at (-cs, 0) with radius C₁ i.e. y² + (x + cs)² = c², where c² = c² - 8². * Using the transversality condition and by differentiating implicitly the equation y²+(x + cs)² = c², show that y' = a/ẞ and cs = y/ß. Finally, show that in the limit as → 0, the stationary path becomes ß²y² + (ßx + y)² = 7².

Sketch the region of integration and change the order of integration. So N f(x, y)dydx S f (x, y) dxdy

Find the line integral of f(x,y) = y e* along the curve r(t) = - 5t i- 12t j, - 1sts0. The integral of f is (Туре an exact answer.)