Practice problems. 1. Evaluate the double or iterated integrals: First: change the order of integration; Second: polar.

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1 Practice problems 1. Evaluate the double or iterated integrals: R x 3 + 1dA where R = {(x, y) : 0 y 1, y x 1}. 1/ 1 y 0 3y sin(x + y )dxdy First: change the order of integration; Second: polar.. Consider the ice-cream cone shaped lamina bounded by x + y = 1, y 0 and y = x 1, x 1. Set up the integrals for area using both dxdy and dydx. 3. Set up the integral for the volume outside x + y = 1 but inside x +y y = 0, below z = y +x and above xy plane. Rewrite it in polar coordinates. If this region has a mass density δ = x + z, set up the integral for the moment of inertia about y axis (don t evaluate). 4. Find the volume bounded by z = 4 x 4 + y 4 +x 1 and z = 4 x 4 + y y. 5. Compute the volume of the solid bounded by z = 1, z = y + 1, z = 1 y, x = y + z and yz plane. Consider the three surfaces that have no x variable first. These three guys form a triangular region in yz plane. For the x direction, it s clearly between x = 0 and x = y + z. You can set it up in triple integral using Cartesian 6. Let T be the region bounded by y = x, x = y, z = 0 and z = x + y. Find the triple integral T xydv. The key is to write out the region: 0 x 1, x y x and 0 z x + y. 7. Let R be the region bounded by ye x = 0, ye x 4 = 0, x 3 6y 1 = 0, x 3 6y = 0 in the first quadrant. Evaluate the double integral I = (e x y + x e x )e x y da. R 8. Let R be the parallelogram with vertices (0, 0), (1, 1), (, 0), (1, 1). Find the integral (x y )da R 1

2 9. Consider the region bounded by y x = 4 (x + y) and y x = 10. If the region has a density δ = (y x). Compute the total mass. Clearly, directly evaluating is not easy because we must discuss different case. We can do change of variables u = x + y, v = y x. 10. Find the area of the region R inside the ellipse x xy +y =. (Hint: try the change of variables x = u 3 v and y = u + 3 v) 11. Consider that T is given by x +y /4+z /9 1. Evaluate the average 1 value of f over T : V olume(t ) T f(x, y, z)dv, where f = x. Do change of variables first u = x, v = y/, w = z/3. Then, the region is transformed into a ball. 1. Consider the region inside the cone with base radius and height, but outside the cylinder with radius 1 that shares the axis with the cone. Assume the mass density is δ = 1. Create a coordinate axis and compute the centroid. convenient in cylindrical coordinates. 13. Consider the the solid outside x + y + (z 1) = 1 but inside ρ = 4 cos φ. Suppose the density is given by δ = x + y. Set up integrals for the centroid and moment of inertia about z axis. Clearly, use spherical coordinates to set up the triple integral. Then, δ = ρ sin φ. 14. Evaluate T xydv where T is the region bounded by x +y x = 0 and x + y + z = 4. What is the volume of this region? Cylindrical coordinates are convenient. The sphere becomes r + z = Set up the integral for T f(x, y, z)dv where f = x + y and T is the region contained in the sphere x + y + (z a) = a but below z = 3 3 r. Spherical coordinates. T : 0 θ < π, π 3 φ π/, 0 ρ a cos φ. f = ρ sin φ. Surface areas:

3 Parametrize the surface y = f(x, z). Use this to compute the area of the plane y = x + z + 1 inside x + z = 1. Set up an integral for the surface area of the surface cut from x = y + 3z z by z + y = 3, z = y. Consider the surface of revolution obtained by revolving x = f(z) about z axis. Parametrize this surface. Consider the fence S: x = sin(t), y = 8 cos(3t), 0 t < π and 0 z. Set up the surface area integral S ds. S is the surface z = θ, 0 θ π and 1 x + y 4. Set up an iterated integral for the surface area. ************************* 1. Divergence, curl (a) In the daytime under sunshine, the algae in an ocean generate oxygen (they also consume oxygen but the net effect is oxygen production). The rate of production is clearly proportional to the density of algae in the ocean. Suppose that the oxygen consumption by other plants and animals in the ocean can be neglected and that the density distribution of the oxygen reaches equilibrium. The equilibrium is kept under a flow of the oxygen which is the result of diffusion, transportation etc. We model the ocean by R 3 and the field of the oxygen flow is given by F = arctan(x)+e y, x 1 + z + x 4 +arctan(y), arctan(z)+sin4 (x) y If the density of the algae at (0, 0, 0) is per cubic centimeter, what is the density at (1, 1, 1)? (b) Suppose there is a cloud of charged dust. generated by the dust is given by E = xy, xyz, sin(x). The electronic field What is the charge density at (0, 0, 0)? If the dust is positively charged at (1, 1, 1) with C/cm 3, what is the charge density at ( 1, 1, )? Is it negatively charged or positively charged? 3

4 (c) In a storm weather, near (π/, 1, 1), the velocity field of the air was roughly given by v = xy, xyz, sin(x). What is the vorticity at (π/, 1, 1)?. Line integrals (a) Parametrization Parametrize x + 4y = 1 r(t) = cos t, 1 sin t, 0 t < π Parametrize the boundary of the region bounded by x-axis, y = x and x = 1. C = C 1 +C +C 3. C 1 : r(t) = t, 0, 0 t 1. C : r(t) = 1, t, 0 t 1. C 3 : r = t, t, t : 1 0 Parametrize the ellipse formed by the intersection of x +y = 1 and x + z = 0. r(t) = cos t, sin t, cos t (b) Usual line integrals ( types) Consider the curve x /4 + y = 1 with x 0, 0 y 1/. If the density (per unit length) is δ = y/x, compute the moment of inertia I y. I y = C x δds = C xyds. r = cos t, sin t. 0 t π/6 Compute the line integral of F = 3y, x over the curve y = x for 0 y 1 oriented from right to left. Let r = t 3, t, t, 0 t 1. Compute C F T ds where F = e yz, 0, ye yz (c) Conservative field. Let C be r(t) = ln(1 + t 9 ), t 3 + 1, t 100, 0 t 1. Compute C xdy + ydx + dz The field is (xy+z). Or you can notice that it is d(xy+z) C F T ds where F = zexz + e x, yz, xe xz + y. r = e t, e t3, t 4. t [0, 1] It is irrotational and thus conservative. φ = e xz + y z Let C be r(t) = cos 4 t, sin 4 t, 7, 0 t < π and F = x 3 z, y 3, y + z 3. Compute the line integral C F d r. (Hint: Split out a conservative field. The integral of the one you split will be zero since it is on a closed curve.) The field F is not conservative but x 3, y 3, z 3 is conservative. Since the curve is closed, we only need to compute 4

5 C z, 0, y dr = C 7dx + ydz = 7 C dx = 0 because 1, 0, 0 is again conservative. 5