REVIEW 3: Key 1. Given the iterated integral ∫ 3 ∫ 9 y sin(x 2) dx ...

REVIEW 3: Key∫ 3 ∫ 91. Given the iterated integral y sin(x 2 ) dx dy, draw the correspondingregion of integration on the xy-plane, reverse the order of integration, and0 y 2evaluate the integral.Solution:y ∫ 3 ∫ 930 y2.5∫ 29 ∫ √ y= x21.510.52 4 6 8x== 1 20 y=0∫ 90y sin(x 2 ) dx dyx sin(x 2 ) dxsin(x 2 ) y dy dx= − 1 4 cos(x2 )∣ 9 0= 1 4(1 − cos 81) ≈ 0.0558.2. Find the volume under the graph of the function f(x, y) = 3xy +y 2 overthe region R defined by the inequalities x ⩾ 0, y ⩾ 0, and x + 2y ⩽ 2.Solution:The region of integration is the triangle bounded by lines x = 0, y = 0, andx + 2y = 2.V ====∫ 2 ∫ y=1−x/20∫ 20∫ 20∫ 20y=0( 32 xy2 + y33 y=0( 32 x(1 − x/2)2 +(3xy + y 2 ) dy dx) y=1−x/2dx(1 −)x/2)3dx3124 (4x3 − 15x 2 + 12x + 4) dx = 2 3 .3. Find the mass of the pyramid bounded by the planes x = 0, y = 0,z = 1, and 3x + 2y + z = 7 given the mass density function δ(x, y, z) = 2z.Key:∫ 2 ∫ 3−3x/2 ∫ 7−3x−2y4. Find the double integral001∫ 3π/4 ∫ 4/ sin θπ/402z dz dy dx = 30.r dr dθ1

in polar coordinates and sketch the region of integration. Interpret the doubleintegral geometrically as an area or volume.Key:∫ 3π/4 ∫ 4/ sin θπ/40r dr dθ =∫ 3π/4π/48sin 2 dθ = 16.θThe region is the triangle bounded by the lines y = x, y = −x, and y = 4.The double integral is the area of the region; it is also the volume of the solidbetween the planes z = 0 and z = 1 over the region.5. Using cylindrical coordinates, find the mass of the upper half (abovethe xy-plane) of the unit ball x 2 + y 2 + z 2 ⩽ 1 given the mass density functionδ(x, y, z) = z. Find the same mass using spherical coordinates.Key:∫ 2π ∫ 100( ∫√ )1−r 2z dz r dr dθ = π ∫ 2π4 =00∫ π/2 ∫ 16. Find parametric equations of(a) the straight line through the points (1, 2, −1) and (2, 3, 4);Key:⎧⎪⎨ x = 1 + ty = 2 + t⎪⎩z = −1 + 5t00ρ cos φ ρ 2 sin φ dρ dφ dθ.(b) the circle of radius 4 centered at the point (−1, 3, 5) and lying in theplane parallel to the xz-plane.Key:⎧⎪⎨ x = −1 + 4 cos ty = 3⎪⎩z = 5 + 4 sin t2