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: x 3 + 1dA where = {(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. The density is δ = 1. Set up the integral for the total mass (which equals the area) in Cartesian coordinates using the order dxdy. (Hint: You need to break the region into two parts.) If I ask you to evaluate the integrals, will you change the coordinates for one of the regions? 3. Set up the integral for the volume inside both x +y = 1 and x +y y = 0, below z = y + x and above xy plane. ewrite 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. Evaluate the volume bounded by the paraboloid z = x + y and the plane z = y. This is the problem that involves the polar coordinates. integral it is (y x y )da In double where is the region bounded by x + y = y. Then, apply polar. This is one example in the lecture notes. This problem can also be evaluated using triple integral by cylindrical coordinates. It s essentially the same as the double integral one. 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 1

2 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 be the region bounded by x + y = 3, y x =, x + y = 1, x y = 4. Compute the double integral (16x 4y )dxdy. We use change of variables u = x + y, v = y x. Then, we compute the Jacobian. 8. Let be the parallelogram with vertices (0, 0), (1, 1), (, 0), (1, 1). Find the integral (x y )da 9. Find the area of the region inside the ellipse x xy +y =. (Hint: try the change of variables x = u 3 v and y = u + 3 v) 10. Let 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. 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. Find the volume under f(x, y) = x and above the region = {(x, y) : (x 1) + y 1, x + (y 1) 1}. convenient in cylindrical coordinates. 13. Consider the ball x +y +z 4. If the density inside x +y +z = 1 is δ 1 = x + y + z while the density outside it is δ = 1. What is the total mass of the ball?

3 14. 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. 15. 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: 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 3

4 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 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? (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 4

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