Download Spring 2003 Mathematics Problems: Definite Integrals and Area Calculation - Prof. J. M. Ma and more Assignments Mathematics in PDF only on Docsity! Spring 2003 Problems Definite Integrals 1. Evaluate the following definite integrals: a. ∫ 3 −1 (2− x+ x2)dx, b. ∫ 4 0 ( x2 + 3− e−x ) dx, c. ∫ 2 0 ( x2 + 1 )2 dx, d. ∫ 3π 0 cos( t2)dt, e. ∫ 5 −1 dx√ 6 + 2x , f. ∫ 5 1 x2 + 1 x dx, g. ∫ π 0 (cos(2t) + 4t) dt, h. ∫ π/2 0 cos(x) 1 + sin(x) dx, i. ∫ 4 3 2x√ 25− x2 dx, j. ∫ π 0 ( 3t 2 − cos(3t) ) dt, k. ∫ 2 −2 ( 1 x+ 3 + e2x ) dx, l. ∫ π/2 0 3 (sin(x) + 1)2 cos(x)dx. 2. Find the area bounded by the function y = 4 − x2 and the x-axis. Sketch the graph of the region. 3. Find the area between the function y = 3 sin(2x) and the x-axis for 0 ≤ x ≤ π/2. Sketch the graph of the region. 4. Consider the curves y = x+ 3 and y = x2 + x− 6. a. Sketch the graph of these curves. Show the x and y-intercepts and any vertices. b. Find the points of intersection of these curves. (Hint: Set the equations equal to each other and solve for x.) c. Find the area between the two curves. 5. Consider the curves y = −2x− 1 and y = 15− 2x− x2. a. Sketch the graph of these curves. Show the x and y-intercepts and any vertices for both curves. b. Find the points of intersection of these curves. (Hint: Set the equations equal to each other and solve for x.) c. Find the area between the two curves. 10. Below are six years of data from some particular animal population (in thousands): Year 0 1 2 3 4 5 6 Pop 53 37 39 54 70 68 52 a. These data are fitted pretty well by the function P (t) = 53− 18 sin (π 3 t ) . Sketch a graph of this curve and show the data points from the table. Use this function to find when the maximum and minimum populations occur and what their values might be. b. Find the average population by computing Pave = 1 6 ∫ 6 0 P (t)dt. 11. One of the hazards of modern medicine is the possible exposure to radioactive sources used in cancer treatment. Radioactive iodine, 131I, is used in the treatment of certain thyroid problems. a. A differential equation describing the radioactive decay of 131I is given by dR dt = −0.1R, R(0) = 50, where t is in days. Solve this differential equation and find the half-life for 131I. b. Suppose a technician is receiving an exposure from mislaid sample of D(t) = 5e−0.1t, in mCi/day. The total exposure over 10 days is given by the integral ∫ 10 0 D(t) dt. Find this total exposure. c. How long can the technician stay near this source if the exposure is to be kept to less than 10 mCi? 12. a. An agricultural pest is experiencing an outbreak that satisfies the Malthusian growth law dP dt = kP, P (0) = 100, where t is in weeks. If it is observed that the population of this pest is 250 after one week, then determine the doubling time of this pest and find a general expression for the population as a function of time. b. The average population over a period of 4 weeks is given by the integral 1 4 ∫ 4 0 P (t) dt. Find the average population of this pest over this period of time. 13. The normal distribution, which is commonly used in statistics, for a random variable x is given by the formula N(x) = 1 σ √ 2π e−x 2/(2σ2), where σ is the standard deviation. The expected value of x over an interval [a, b] is given by the formula ∫ b a xN(x) dx. This can be interpreted as the average value of x that one expects over the interval. Find the expected value of x if σ = 1 and the interval is [0, 2].
