MATH 251 Fall 2016 EXAM III - VERSION A

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1 MATH 51 Fall 16 EXAM III - VERSION A LAST NAME: FIRST NAME: SECTION NUMBER: UIN: DIRECTIONS: 1. You may use a calculator on this exam.. TURN OFF cell phones and put them away. If a cell phone is seen during the exam, your exam will be collected and you will receive a zero. 3. In Part 1 (Problems 1-8), mark the correct choice on your ScanTron using a No. pencil. The ScanTron will not be returned, therefore for your own records, also record your choices on your exam! Each problem is worth 4 points. 4. In Part (Problems 9-14), present your solutions in the space provided. Show all your work neatly and concisely and clearly indicate your final answer. You will be graded not merely on the final answer, but also on the quality and correctness of the work leading up to it. 5. Be sure to write your name, section number and version letter of the exam on the ScanTron form. THE AGGIE CODE OF HONOR An Aggie does not lie, cheat or steal, or tolerate those who do. Signature: DO NOT WRITE BELOW! Question Type Points Awarded Points Multiple Choice 4 Free Response 6 Total 1 1

2 PART I: Multiple Choice. 5 points each. 1. Which double integral gives the volume under the paraboloid z = x +y and above the curve r = 1 cosθ? (a) (b) (c) (d) (e) π 1 cosθ π 1 cosθ π 1 cosθ π 1 cosθ π 1 cosθ r 3 drdθ r drdθ rdrdθ r 3 drdθ rdrdθ. Find the volume under the surface z = 4x and above the triangle with vertices (,), (1,) and (,4). (a) 14 3 (b) 8 (c) 16 3 (d) 16 (e) 8 3

3 3. Find the volume of the solid in the first octant inside the cylinder x +y = 4 and below the paraboloid z = x +y. (a) π (b) π (c) 4π (d) π (e) π 4 4. Compute (a) 81 π (b) 7π (c) 43π (d) 7π 4 (e) 34π 5 E x +y dv where E is region bounded by the paraboloid z = 9 x y and the xy plane. 3

4 5. Reverse the order of integration for 1 y y f(x,y)dxdy. (a) (b) (c) (d) (e) 4 x x 4 x + + x + 4 x 1 x 1 4 x 1 x 6. Consider E xydv, where E is the region bounded by the parabolic cylinders x = y and y = x and the planes z = and z = 3x+y. If we choose dv = dzdydx, which of the following is the resulting double integral? (a) (b) (c) (d) (e) x 1 x x x (6x y +x y )dydx x (6x y +x y )dydx (6x y +xy )dydx x (6x y +xy )dydx (6x y +xy )dydx 4

5 7. Reverse the order of integration for (a) (b) (c) (d) (e) lnx 1 ln ln e y 1 ln e y ln 1 ln e y e y f(x,y)dxdy f(x,y)dxdy f(x,y)dxdy e y f(x,y)dxdy f(x,y)dxdy. 8. Find the area of the region inside the circle r = 4sinθ and outside the circle r = (a) π 3 3 (b) 3+ 4π 3 (c) 3+ π 3 (d) 4 3+ π 3 (e) 3 4π 3 5

6 Part II: Work out. Show all intermediate steps. Special note: For each integral, you must have its corresponding differential (ie dx, dy,dz, dθ, dφ, dρ). 9. (8 pts) Evaluate 4 π π y ysin(x )dxdy. Show all steps. Simplify your answer. 1. (8 pts) The solid E is the region bounded below by the cone φ = π 6 Set up xyzdv. Collect like terms, but do not integrate. E and above by the sphere ρ =. 6

7 4 11. (8 pts) Rewrite 4 16 x 16 x like terms, but do not integrate. 16 x y z x +y +z dzdydx by converting to spherical coordinates. Collect 1. (8 pts) Consider the region E bounded by z = 4 3x 3y and z = x +y. Set up (7xyz) dv. Collect like terms, but do not integrate. E 7

8 13. For the following, r and θ π. (a) Convert the rectangular point ( 3,1, 3) to: i. (3 pts) cylindrical ii. (3 pts) spherical (b) Convert the cylindrical point (6, π ) 3,5 to: i. (3 pts) rectangular ii. (3 pts) spherical (c) Convert the spherical point i. (3 pts) rectangular (, 3π 4, π ) to: 6 ii. (3 pts) cylindrical 8

9 14. (1 pts) Conider the solid tetrahedron with vertices (,,), (,,), (,,) and (,,4). Set up but do not evaluate an integral that gives the volume of the solid using the order dv = dzdydx. 9