Download test 3 problems

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Mathematics 152 third test, parts one, two, and three
due Wednesday, June 1, 2011
please show relevant work to get full credit for each problem
1. Consider the unbounded second quadrant region R bounded by the negative x-axis, the
y-axis, and y = ex :
R = {(x, y) | 0 ≤ y ≤ ex and x ≤ 0 }
Set up integrals for, and evaluate (a) the area of R, (b) the x and y coordinates of the center
of mass of R, (c) the volume obtained if R is rotated around the x-axis, (d) the volume
obtained if R is rotated around the line y = 1 − x.
2. Walk one unit along the parabola y = x2 from the vertex. Find the x and y coordinates
of the points that you get to. (see the auxiliary graph to finish the problem!)
3. Consider the region H in the first and fourth quadrants bounded on the right by the unit
hyperbola x2 − y 2 = 1 (x > 0), bounded above by the line segment from the origin to the
point (a, b) , and bounded below by the line segment from the origin to the point (a, −b) .
( a > 0 and b > 0)
(a) Calculate the area of H using vertical rectangles and a trigonometric substitution
(b) Calculate the area of H using vertical rectangles and a hyperbolic trigonometric substitution
(c) Calculate the area of H using horizontal rectangles and a trigonometric substitution
(d) Calculate the area of H using horizontal rectangles and a hyperbolic trigonometric
substitution
4. Set up, but do not bother to evaluate, the definite integral for the surface area of the
surface obtained by rotating the curve y = x3 , 0 ≤ x ≤ 2, around the vertical line x = 6.
5. Suppose that the amount of money P owed on a housing loan at time t years satisfies
the differential equation
dP
= .05P − k
dt
(k a constant)
(a) Solve the differential equation, obtaining a solution for P involving the constant k
(b) What is a constant solution for the differential equation?
(c) Calculate a formula for the specific solution satisfying the initial condition P (0) =
250000 and P (30) = 0.
(d) For the solution of part (c), at what time t is P (t) = 125000 ?
continues on next page
6. Is the function y =
Z x
cos t2 dt a solution of
2
dy
= cos x2 and y(2) = 0 ? Explain.
dx
7. Find the curves which are orthogonal to the cubic curves y = x3 + C
8. Obtain an approximation for the definite integral of problem (4) by using Simpson’s rule
with n = 10.
9. Given the logistic differential equation
dP
= (2×10−2 )P −(2×10−11 )P 2
dt
(a) Find all constant solutions
(b) Sketch the direction field
(showing the constant solutions, and showing where the slopes are zero,
positive, or negative)
(c) Find the solution corresponding to initial condition P (0) = 2 × 107 .
(d) For the solution of part (c), at what value of t does P (t) = 5 × 108 ?
(e) For the solution of part (c), what is the limit of P (t) as t → ∞ ?