526 CHAPTER 26. THE RIEMANNN INTEGRAL ON Rp

the graph of 13 x+ y = 2 on the top. Therefore, in setting up the integrals, with the integral

with respect to x on the outside, the double integral equals the following sum of iteratedintegrals.

has x=3y2 on top︷ ︸︸ ︷∫ 3

0

∫ √x/3

x/9ydydx+

has 13 x+y=2 on top︷ ︸︸ ︷∫ 9

2

3

∫ 2− 13 x

x/9ydydx

You notice it is not necessary to have a perfect picture, just one which is good enough tofigure out what the limits should be. The dividing line between the two cases is x = 3 andthis was shown in the picture. Now it is only a matter of evaluating the iterated integralswhich in this case is routine and gives 1.

26.2 Exercises1. Evaluate the iterated integral and then write the iterated integral with the order of

integration reversed.∫ 4

0∫ 3y

0 xdxdy.

2. Evaluate the iterated integral and then write the iterated integral with the order ofintegration reversed.

∫ 30∫ 3y

0 ydxdy.

3. Evaluate the iterated integral and then write the iterated integral with the order ofintegration reversed.

∫ 20∫ 2y

0 (x+1)dxdy.

4. Evaluate the iterated integral and then write the iterated integral with the order ofintegration reversed.

∫ 30∫ y

0 sin(x) dxdy.

5. Evaluate the iterated integral and then write the iterated integral with the order ofintegration reversed.

∫ 10∫ y

0 exp(y) dxdy.

6. Let ρ (x,y) denote the density of the plane region closest to (0,0) which is betweenthe curves x+2y = 3,x = y2, and x = 0. Find the total mass if ρ (x,y) = y. Set up theintegral in terms of dxdy and in terms of dydx.

7. Let ρ (x,y) denote the density of the plane region determined by the curves x+2y =3,x = y2, and x = 4y. Find the total mass if ρ (x,y) = x. Set up the integral in termsof dxdy and dydx.

8. Let ρ (x,y) denote the density of the plane region determined by the curves y =2x,y = x,x+ y = 3. Find the total mass if ρ (x,y) = y+ 1. Set up the integrals interms of dxdy and dydx.

9. Let ρ (x,y) denote the density of the plane region determined by the curves y =3x,y = x,2x+ y = 4. Find the total mass if ρ (x,y) = 1.

10. Let ρ (x,y) denote the density of the plane region determined by the curves y =3x,y = x,x+ y = 2. Find the total mass if ρ (x,y) = x+ 1. Set up the integrals interms of dxdy and dydx.

11. Let ρ (x,y) denote the density of the plane region determined by the curves y =5x,y = x,5x+ 2y = 10. Find the total mass if ρ (x,y) = 1. Set up the integrals interms of dxdy and dydx.