Engineering Math

14.2 Exercises

1. Evaluate the iterated integral and then write the iterated integral with the order of integration reversed. ∫ ₀⁴ ∫ ₀^3yxdxdy.
2. Evaluate the iterated integral and then write the iterated integral with the order of integration reversed. ∫ ₀³ ∫ ₀^3yydxdy.
3. Evaluate the iterated integral and then write the iterated integral with the order of integration reversed. ∫ ₀² ∫ ₀^2y
   (x+ 1)
   dxdy.
4. Evaluate the iterated integral and then write the iterated integral with the order of integration reversed. ∫ ₀³ ∫ ₀^y sin
   (x)
   dxdy.
5. Evaluate the iterated integral and then write the iterated integral with the order of integration reversed. ∫ ₀¹ ∫ ₀^y exp
   (y)
   dxdy.
6. Let ρ
   (x,y)
   denote the density of the plane region closest to
   (0,0)
   which is between the curves x + 2y = 3,x = y², and x = 0. Find the total mass if ρ
   (x,y)
   = y. Set up the integral 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 terms of 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 in terms 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 in terms 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 in terms of dxdy and dydx.
12. Find ∫ ₀⁴ ∫ _y∕2²
    1 x
    e2
    yx
    dxdy. You might need to interchange the order of integration.
13. Find ∫ ₀⁸ ∫ _y∕2⁴
    1x
    e3
    y x
    dxdy.
14. Find ∫ ₀
    13
    π ∫ _x
    13
    π
    siny y
    dydx.
15. Find ∫ ₀
    1 2
    π ∫ _x
    1 2
    π
    sinyy
    dydx.
16. Find ∫ ₀^π ∫ _x^π
    siny y
    dydx
17. ^∗ Evaluate the iterated integral and then write the iterated integral with the order of integration reversed. ∫ ₋₃³ ∫ _−x^xx²dydx

    Your answer for the iterated integral should be ∫ ₃⁰ ∫ ₋₃^−yx²dxdy + ∫ ₀⁻³ ∫ ₋₃^yx²dxdy + ∫ ₀³ ∫ _y³x²dxdy + ∫ ₋₃⁰ ∫ _−y³x²dxdy. This is a very interesting example which shows that iterated integrals have a life of their own, not just as a method for evaluating double integrals.

18. ^∗ Evaluate the iterated integral and then write the iterated integral with the order of integration reversed. ∫ ₋₂² ∫ _−x^xx²dydx.