CARNEGIE MELLON UNIVERSITY
DEPARTMENT OF MATHEMATICAL SCIENCES
21-122 Review Exam 1, Spring , 2006

  1. Integrate and simplify your result.

    \begin{displaymath} \int_{0}^{\pi/4} \cos^2 t \sin^4 t \ dt \end{displaymath}

  2. Integrate

    \begin{displaymath} \int \frac{dx}{x^2\sqrt{16-x^2} } \end{displaymath}

  3. Integrate

    \begin{displaymath} \int \frac{1}{x^2(x^2+1)} \ dx \end{displaymath}

  4. Determine if the following integral converges or diverges, and evaluate the integral if it converges.

    \begin{displaymath} \int_{0}^{\infty} \sin x e^{-x} \ dx \end{displaymath}

  5. The rate that water is leaking from a tank is given by the following table:

    time (hours) 0.0 2.0 4.0 6.0 8.0
    rate (liters/hours) 12 6 3 1 0

    1. Approximate the total amount of water that leaked from the tank in these 8 hours using the Trapezoid Rule.

    2. Suppose that the volume, $V$, of water in the tank at time $t$ satisfies $-0.6 \geq d^2V/dt^2 \geq -6 $ liters/hour${}^2$, and $0.1 \leq d^3V/dt^3 \leq 1$ liters/hour${}^3$. Determine an upper bound for the maximum error in your approximation using the Trapezoid Rule.

    3. Determine, with explanation, whether your approximation using the Trapezoid Rule is an overestimate or an underestimate.

    4. Approximate the total amount of water that leaked from the tank in these 8 hours using Simpson's Rule.



Timothy J Flaherty 2006-02-09