A partially silvered mirror covers the square area with vertices at (1, 1). The fraction

Z 2 sin cos d = sin2 or cos2 or 1 2 cos 2. Hint: Use trig identities

Z dx x2 + a2 = sinh1 x a or ln x + px2 + a2 . Hint: To find the sinh1 form,

Z dy py2 a2 = cosh1 y a or ln y + py2 a2 . Hint: See Problem 2 hints.

Z p1 + a2x2 dx = x 2 p1 + a2x2 + 1 2a sinh1 ax or x 2 p1 + a2x2 + 1 2a ln ax + p1 + a2x2 .

Z K dr 1 K2r2 = sin1 Kr or cos1 Kr or tan1 Kr 1 K2r2 .

Z K dr r r2 K2 = cos1 K r or sec1 r K or sin1 K r or tan1 K r2 K2 .

RR A(2x 3y) dx dy, where A is the triangle with vertices (0, 0), (2, 1), (2, 0).

RR A 6y2 cos x dx dy, where A is the area inclosed by the curves y = sin x, the x axis, and the line x = /2.

In Problems 7 to 18 evaluate the double integrals over the areas described. To find the limits, sketch the area and compare Figures 2.5 to 2.7.

RR A y dx dy where A is the area in Figure 2.8.

RR A x dx dy, where A is the area between the parabola y = x2 and the straight line 2x y + 8 = 0.

RR y dx dy over the triangle with vertices (1, 0), (0, 2), and (2, 0).

RR 2xy dx dy over the triangle with vertices (0, 0), (2, 1), (3, 0).

RR x2ex2y dx dy over the area bounded by y = x1, y = x2, and x = ln 4.

RR dx dy over the area bounded by y = ln x, y = e + 1 x, and the x axis

RR (9 + 2y2) 1 dx dy over the quadrilateral with vertices (1, 3), (3, 3), (2, 6), (6, 6).

RR (x/y) dx dy over the triangle with vertices (0, 0), (1, 1), (1, 2).

RR y1/2 dx dy over the area bounded by y = x2, x + y = 2, and the y axis

Above the square with vertices at (0, 0), (2, 0), (0, 2), and (2, 2), and under the plane z = 8 x + y.

Above the rectangle with vertices (0, 0), (0, 1), (2, 0), and (2, 1), and below the surface z2 = 36x2(4 x2).

Above the triangle with vertices (0, 0), (2, 0), and (2, 1), and below the paraboloid z = 24 x2 y2.

Above the triangle with vertices (0, 2), (1, 1), and (2, 2), and under the surface z = xy.

Under the surface z = y(x + 2), and over the area bounded by x + y = 0, y = 1, y = x.

Under the surface z = 1/(y+2), and over the area bounded by y = x and y2+x = 2.

Z 1 x=0 Z 33x y=0 dy dx

Z 2 y=0 Z 1 x=y/2 (x + y) dx dy

Z 4 x=0 Z x y=0 y x dy dx

1 y=0 Z 1y2 x=0 y dx dy

Z y=0 Z x=y sin x x dx dy

2 x=0 Z 2 y=x ey2/2 dy dx

Z ln 16 x=0 Z 4 y=ex/2 dy dx ln y

Z 1 y=0 Z 1 x=y2 ex x dx dy

A lamina covering the quarter disk x2 + y2 4, x > 0, y > 0, has (area) density x + y. Find the mass of the lamina.

A dielectric lamina with charge density proportional to y covers the area between the parabola y = 16 x2 and the x axis. Find the total charge.

A triangular lamina is bounded by the coordinate axes and the line x + y = 6. Find its mass if its density at each point P is proportional to the square of the distance from the origin to P.

A partially silvered mirror covers the square area with vertices at (1, 1). The fraction of incident light which it reflects at (x, y) is (xy) 2/4. Assuming a uniform intensity of incident light, find the fraction reflected.

Z 2 x=1 Z 2x y=x Z yx z=0 dz dy dx

Z 2 z=0 Z 2 x=z Z z y=8x dy dx dz

Z 3 y=2 Z 2 z=1 Z 2y+z x=y+z 6y dx dz dy

Z 2 x=1 Z 2x z=x Z 1/z y=0 z dy dz dx.

Find the volume between the planes z = 2x + 3y + 6 and z = 2x + 7y + 8, and over the triangle with vertices (0, 0), (3, 0), and (2, 1).

Find the volume between the planes z = 2x + 3y + 6 and z = 2x + 7y + 8, and over the square in the (x, y) plane with vertices (0, 0), (1, 0), (0, 1), (1, 1).

Find the volume between the surfaces z = 2x2 + y2 + 12 and z = x2 + y2 + 8, and over the triangle with vertices (0, 0), (1, 0), and (1, 2).

Find the mass of the solid in Problem 42 if the density is proportional to y.

Find the mass of the solid in Problem 43 if the density is proportional to x.

Find the mass of a cube of side 2 if the density is proportional to the square of the distance from the center of the cube.

Find the volume in the first octant bounded by the coordinate planes and the plane x + 2y + z = 4.

Find the volume in the first octant bounded by the cone z2 = x2 y2 and the plane x = 4.

Find the volume in the first octant bounded by the paraboloid z = 1 x2 y2, the plane x + y = 1, and all three coordinate planes.

Find the mass of the solid in Problem 48 if the density is z.