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Course Notes for Calculus II

Remkes Kooistra

Section 13.5 Week 7 Assignment

  1. Calculate the following integral. Shows your steps and reasoning. (4)
    \begin{gather*} \int_1^{\infty} \frac{\ln x}{x^2} dx \end{gather*}
  2. Calcluate the slope of this parametric curve. Identity any vertical tangents by finding places where the slope approaches infinity. (4)
    \begin{gather*} \gamma(t) = (3t^2 - t - 1, t^2 + 2t + 4) \end{gather*}
  3. Calcluate the slope of this parametric curve. Identity any vertical tangents by finding places where the slope approaches infinity. (4)
    \begin{gather*} \gamma(t) = \left( \frac{1}{1-t^2}, \frac{t}{1-t^2} \right) \end{gather*}
  4. Calculate the length of this parametric curve. (4)
    \begin{align*} \amp \gamma(t) = (5t - 1, 3t-2) \amp \amp t \in [0,10] \end{align*}
  5. Calculate the length of this parametric curve. (4)
    \begin{align*} \amp \gamma(t) = (9t,t^2) \amp \amp t \in [0,2] \end{align*}
    For the resulting integral, you can use this antiderivative instead of doing all of the work of a trig substitution.
    \begin{equation*} \int \sqrt{a^2 + b^2t^2} dt = \frac{t}{2} \sqrt{a^2 + b^2t^2} + \frac{a^2}{b^2} \ln \left| \frac{bt}{a} + \sqrt{\frac{b^2t^2}{a^2} + 1} \right| + c \end{equation*}
    The result you will get from this integral will be a complicated expression with square roots and logarithms. Leave your answer in this form; do not change it into decimals.
  6. Consider these two parametric curves. (8)
    \begin{align*} \amp \gamma(t) = \left( \frac{\cos t}{t}, \frac{\sin t}{t} \right) \amp \amp t \in [0,\infty)\\ \amp \gamma(t) = \left( \frac{\cos t}{t^2}, \frac{\sin t}{t^2} \right) \amp \amp t \in [0,\infty) \end{align*}
    Both of these are inward spirals, but they spiral inwards at different rates. Prove that \(\gamma_1\) has infinite length but \(\gamma_2\) has finite length.
    A couple hints for this questions: First, recall from the section on improper integrals that convergence of integrals is detemined by the asymptotic order of the integrand. For these, you need to setup the length integral for each curve, but you don’t actually need to do that calculation. Instead, find the asymptotic order of each integrand and use the examples from the section on improper integrals to determine whether or not they converge: if the integral converges, the length is finite and if it diverges, the length is infinite. Second, when you are calculating \(\sqrt{x^\prime(t)^2 + y^\prime(t)^2}\) for each of these, expect long expressions with lots of trig terms. However, using trig identities, these expression will collapse down into much more concise expressions without any trig functions left.
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