Homework 10

Directions:

  1. Show each step of your work and fully simplify each expression.
  2. Turn in your answers in class on a physical piece of paper.
  3. Staple multiple sheets together.
  4. Feel free to use Desmos for graphing.

Answer the following:

  1. Why can't you use FTC to find this integral? \[\int^2_{-2} \dfrac{1}{x} \ dx\]
  2. Suppose $f(x)$ is differentiable on $(-\infty, 3) \cup (3, \infty)$. Can you use FTC to evaluate \[\int^{10}_{-10} f(x) \ dx\] Why or why not?
  3. Suppose \[\int f(x) \ dx = F(x) + C\] What should $\dfrac{d}{dx} \left[F(x) + C\right]$ be equal to?
  4. Is this true? \[\int \tan^2 x \ dx = \tan x - x + C\]
    Hint: If you get stuck, use the identity $\tan^2x + 1 = \sec^2x$ and solve for $\tan^2 x$.
  5. What is the piecewise definition of $f(x) = \lvert x \rvert$?
  6. Find the following integrals:
    1. $\displaystyle \int \sqrt[4]{x^5} \ dx$
    2. $\displaystyle \int \sec \theta(\sec \theta + \tan \theta) \ d\theta$
    3. $\displaystyle \int \left(u^2 - \dfrac{1}{u^2}\right) \ du$
    4. $\displaystyle \int^6_0 \lvert{x - 3}\rvert \ dx $
    5. $\displaystyle \int^2_{-2} \lvert x^2 - 1 \rvert \ dx $
  7. If $z'(\theta)$ is some rate of change, what should \[\int^8_{4} z'(\theta)\ d\theta\] be?
  8. If $f(t)$ is the rate at which you complete your Calculus homework (in problems finished per minute), what does the integral \[\int^{30}_{0} f(t) \ dt\] mean in English?
  9. What is the difference between displacement and distance?
  10. A particle is moving on a horizontal line. It's velocity at time $t$ is given by $v(t) = t^2 - 2t - 3$.
    1. Calculate the total displacement on the time interval $[2, 4]$.
    2. Calculate the total distance on the time interval $[2, 4]$.
  11. A particle is moving on a horizontal line. It's velocity at time $t$ is given by $v(t) = 3t - 6$.
    1. Calculate the total displacement on the time interval $[0, 3]$.
    2. Calculate the total distance on the time interval $[0, 3]$.
  12. Describe in English the high-level concept underpinning U-Substitution (not the algebraic nitty-gritty details).
  13. Consider the expression \[\int 2x \sqrt{1 + x^2} \ dx \]
    1. In Section 4.2, what did we say the $dx$ represents?
    2. In Section 4.5, what can we treat the $dx$ as? Why can we do this?
  14. Describe the algebraic differences between using U-Substitution on definite and indefinite integrals.
  15. True or False: \[\int x^2 \sqrt{x^3 + 1} \ dx = \left(\int x^2 \ dx\right) \cdot \left(\int \sqrt{x^3 + 1} \ dx\right)\]
  16. Evaluate the following integrals:
    1. $\displaystyle \int x^2 \sqrt{x^3 + 1} \ dx$
    2. $\displaystyle \int^{1}_0 x^2\sqrt{x + 1}\ dx$
    3. $\displaystyle \int \sin^2 \theta \cos \theta \ d\theta$
      Hint: Choose $u = \sin \theta$.
    4. $\displaystyle \int^{\sqrt{\pi}}_0 x\cos(x^2)\ dx$
    5. $\displaystyle \int \sin \left(\dfrac{3x}{4}\right) \ dx$
    6. $\displaystyle \int^{2}_1 x\sqrt{x - 1}\ dx$
    7. $\displaystyle \int \sqrt{\cot x} \csc^2 x \ dx$
    8. $\displaystyle \int^{\pi / 2}_0 \cos(x)\sin(\sin(x)) \ dx$