19.3. METHODS FOR TRIPLE INTEGRALS 355
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.
12. Find∫ 4
0∫ 2
y/21x e2 y
x dxdy. You might need to interchange the order of integration.
13. Find∫ 8
0∫ 4
y/21x e3 y
x dxdy.
14. Find∫ 1
3 π
0∫ 1
3 π
xsiny
y dydx.
15. Find∫ 1
2 π
0∫ 1
2 π
xsiny
y dydx.
16. Find∫
π
0∫
π
xsiny
y dydx
17. ∗ Evaluate the iterated integral and then write the iterated integral with the order ofintegration reversed.
∫ 3−3∫ x−x x2 dydx
Your answer for the iterated integral should be∫ 0
3∫ −y−3 x2 dxdy+
∫ −30∫ y−3 x2 dxdy+∫ 3
0∫ 3
y x2 dxdy+∫ 0−3∫ 3−y x2 dxdy. This is a very interesting example which shows that
iterated integrals have a life of their own, not just as a method for evaluating doubleintegrals.
18. ∗ Evaluate the iterated integral and then write the iterated integral with the order ofintegration reversed.
∫ 2−2∫ x−x x2 dydx.
19.3 Methods For Triple Integrals
19.3.1 Definition Of The IntegralThe integral of a function of three variables is similar to the integral of a function of twovariables. In this case, the term: “mesh” refers to a collection of little boxes which coversa given region in R.