Math 215Homework Set 7: §§15.8–16.1Winter 2012Most of the following problems are modified versions of the problems from your text book, MultivariableCalculus, 7th ed., by James Stewart. Your solution to each problem should be complete, show all work,and be written in complete sentences where appropriate. For Maple problems, include a print-out thatshows all of the work and graphs that you generated in Maple to solve the problem, in addition to anywork you may have done by hand.15.8.1: Sketch the solid described by the inequalities (in cylindrical coordinates) −π/4 ≤ θ ≤ π/4, r ≤z ≤ 2.15.8.2: Sketch the solid whose volume is given by the integralRπ/20R20R9−r20r dz dr dθ. Then evaluate theintegral to find the volume.15.9.1: Sketch the solid described by the inequalities (in spherical coordinates) ρ ≤ 2, ρ ≤ csc φ.15.9.2: Consider the surface given in spherical coordinates by the equationρ = cos φ.Use Maple to graph this surface. Then find the volume enclosed by this surface.15.9.3: Show thatZ∞0Z∞0Z∞0px2+ y2+ z2e−(x2+y2+z2)dx dy dz =π4.(The improper integral is defined as the limit of a triple integral over the piece of a solid spherewhich lies in the first octant as the radius of the sphere increases indefinitely).16.1.1: Match the vector fields F on R3with the plots on the next page labeled I-IV (This is problems#15–18 in §16.1). In each case, explain your answers.(a) F(x, y, z) = i + 2 j + 3 k(b) F(x, y, z) = i + 2 j + z k(c) F(x, y, z) = x i + y j + 3 k(d) F(x, y, z) = x i + y j + z kI IIIIIIV16.1.2: Consider the vector field F(x, y) = x i + j.(a) Sketch F for 0 ≤ x ≤ 3, 0 ≤ y ≤ 3, including vectors at integer points in that domain.(b) Sketch flow lines starting at (1/2, 0), (1, 0), and (2, 0). What shape do they have? What typeof function do you think describes them?(c) Show given the definition of F that the slope of the tangent line to any flow line at the point(x, y) is given bydydx=1x.(d) Solve this differential equation (by simply taking an antiderivative) to find the equation of theflow line that goes through the point (1,0). Is this consistent with your guess in