Problems

In Problems 1-16, evaluate the double integrals (compare these with the problems from Section 12.1).

In Problems 17-24, evaluate the iterated integral. Then check your answer, by evaluating in the other order.

In Problems 25-30 evaluate the iterated integral.

In Problems 31-38, find inequalities which describe the given region D, and write down an iterated integral equal to ∫∫_D f(x, y) dA.

31            The triangle with vertices (0, 0), (5, 0), (0, 5).

32            The triangle with vertices (1, - 2), (1, 4), (5, 0).

33            The circle of radius 2 with center at the origin.

34            The bottom half of the circle of radius 1 with center at (2, 3).

35            The region bounded by the parabola y = 4 - x² and the line y = 3x.

36            The region above the parabola y = x² and inside the circle x² + y² = 1.

37            The region bounded by the curves x = ½ and x = 1/(1 + y²).

38            The region bounded by the curves x = 12 + y² and x = y⁴.

39            Find the volume of the solid over the region x² + y² ≤ 1 and between the surfaces z = 0, z = x².

40            Find the volume of the solid over the region D:1 ≤ x ≤2,x ≤ y ≤ x² and between the surfaces z = 0, z = y/x.

41             Find the volume of the solid between the surfaces z = 0, z = 2 + 3x - y, over the region 0 ≤ x ≤ 2, 0≤y≤x

42            Find the volume of the solid between the surfaces z = 0, z =, over the region 0≤x≤l,x≤y≤l.

43            Find the volume of the solid bounded by the plane z = 0 and the paraboloid

44            Find the volume of the solid bounded by the three coordinate planes and the plane ax + by + cz = 1, where a, b, and c are positive.

45            Show that

46            Show that

47            Show that

48            Show that

49            Let

Show that:

1. 
2. For each constant y₀ ≠ ½, the function g(x) = f(x, y₀) is everywhere discontinuous, so that the iterated integral ∫₀¹∫₀¹ f(x, y) dx dy is undefined.