8. Set up the integral to determine the force on the side of a fish tank that has a length of 4 ft and a heght of 2 ft if the tank is full.

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1 . Determine the volume of the solid formed by rotating the region bounded by y = 2 and y = 2 for 2 about the -ais. 2. Determine the volume of the solid formed by rotating the region bounded by the -ais and the curve y = 2e for 2 about the line y =. 3. Determine the volume of the solid formed by rotating the region bounded by the y-ais, the -ais, and the curve y = cos for π about the line =. 2. Determine the volume of the solid formed by rotating the region bounded by the y-ais, the line y =, and the curve y = + 2 for about the y-ais. 5. Set up the integral (do not evaluate) to determine the work to empty a full tank of water that is the shape of a right circular cone having bottom radius ft and top radius 6 ft and height 8 ft. Water has density approimately 62. lb/ft 3. (See picture) Set up the integral (do not evaluate) to determine the work to empty a full swimming pool of water that has a rectangular bottom with length 2 ft and width ft and rectangular top with length 2 ft and width ft. The pool is 8 ft underground. Water has density approimately 62. lb/ft 3. (See picture) 5 7. Set up the integral (do not evaluate) to determine the force on the trapezoidal side of the pool that is shown above when the pool is full of water.

2 8. Set up the integral to determine the force on the side of a fish tank that has a length of ft and a heght of 2 ft if the tank is full. 9. Set up the integral to determine the arclength of the curve y = on 2 (do not evaluate).. Set up the integral to determine the arclength of the curve y = sin (2) on π 2 (do not evaluate).. Determine the average value of the function f() = 3 on Determine the average value of the function f() = on Evaluate the integral if it converges. Show divergence otherwise. (b) (c) 3e 2 d d ( ) 2 d (d) 2 d. Solve the initial value differential equation for y(t): dy dt = ty3 y() = 5. Solve the initial value differential equation for y(t): dy dt = t y y() = 5 6. Use Euler s Method to approimate y() if dy d = y + with y() = 2 and = 2.

3 7. Use Euler s Method to approimate y() if dy d = 2 y with y() = and = A tank contains L of pure water. Brine containing.5 kg of salt per liter of water enters the tank at a rate of 5 liters per minute. Brine that contains. kg of salt per liter of water enters the tank at a rate of liters per minute. The solution is kept thoroghly mied and drains at a rate of 5 liters per minute. Determine a differential equation involving the amount of salt in the tank at a given time, t. (b) Solve the differential equation giving the amount of salt in the tank at any time, t. 9. A tank contains 8 L of pure water. Brine containing. kg of salt per liter of water enters the tank at a rate of 5 liters per minute. Brine that contains.2 kg of salt per liter of water enters the tank at a rate of liters per minute. The solution is kept thoroughly mied and drains at a rate of 5 liters per minute. Determine a differential equation involving the amount of salt in the tank at a given time, t. (b) Solve the differential equation giving the amount of salt in the tank at any time, t. 2. Solve the initial value first order linear differential equation: y = 2y + y() =. 2. Give the general solution to the second order differential equation. y + 2y + 5y = 22. Give the solution to the initial value second order nonhomogeneous differential equation: y + 2y + y = sin y() = y () =. 23. Tell whether the series converges or diverges and justify your answer by showing reason by a valid test. n= n 2 +

4 (b) n= n 2 2 n 2. Determine the given sum: n= ( ) n 3 n n! = (b) = 25. Determine the interval and radius of convergence of the given series: n= ( ) n n n2 n 26. Determine the Taylor Series about = for: f() = 2 e (b) g() = (c) h() = arctan (3) Use the binomial epansion to determine the first 3 terms of the MacLaurin series representing the funtion: f() = Write out the first three NONZERO terms to the MacLaurin series of f() = e cos 29. Use series to approimate e d. (Write out terms but do not add them.) 3. If a = 3, 2, 5 and b = 6, 2, 3, calculate: a + b, a, a b, a b, and cosθ=angle between a and b. (b) Π, the equation of the plane parallel to both a and b containing the point P(, 9, 2).

5 (c) P, the projection vector of a onto b. 3. Change from rectangular coordinates to cylindrical coordinates: P(, 3, 2) Q(,, 8) z y 2 = 9 z = 2 y Change from rectangular coordinates to spherical coordinates: P(, 3, 2 3) Q(7, 7, ) z = 2 + y y 2 + z 2 + 2z = Change from spherical coordinates to rectangular coordinates: P(,, π/) Q(3, π/3, 5π/6) ρ = 2 cosφ φ = π/6 3. Sketch the graph of the given surface and identify it. f(, y) = 6 2 3y (b) f(, y) = 2 + y 2 (c) y 2 + z 2 + z + = (d) f(, y) = 6 2 y 2 (e) 2 + 3y 2 = 9