The Fundamental Theorem
of Calculus, and Integration

  1. Evaluate these indefinite integrals. Don’t forget the \(+C\)s!

    \(\displaystyle\int \csc^2\bigl(2x\bigr) \,\mathrm{d}x\)
    \(\displaystyle\int \frac{\sin(\theta)\,\mathrm{d}\theta}{\sqrt{\cos(\theta)}}\)
    \(\displaystyle\int t\sqrt{5t+6} \,\mathrm{d}t\)

  2. What are the exact values of these definite integrals?

    \(\displaystyle\int\limits_{\pi/8}^{\pi/6} \csc^2\bigl(2x\bigr) \,\mathrm{d}x\)
    \(\displaystyle\int\limits_{\pi/2}^{\pi/3} \frac{\sin(\theta)\,\mathrm{d}\theta}{\sqrt{\cos(\theta)}} \)
    \(\displaystyle\int\limits_{-6/5}^{19/5} t\sqrt{5t+6} \,\mathrm{d}t\)
  3. What is the average value of \(f(x) = x\sqrt{5x+6}\) on the interval \(\bigl[-\tfrac{6}{5}, \tfrac{19}{5}\bigr]?\)

  4. You are standing on the shoulder of a straight Nevada highway, a radar gun in your hand, and a Maserati cruising on the highway headed towards you. You point the radar gun towards the car to measure its speed and, noticing you, the driver of the Maserati floors it. The car speeds towards you, past you, and then disappears into the horizon. The radar gun is equipped with a small computer, and it reports a model \({f(t) \!=\! 120 \!+\! 2t^2 \!-\! \tfrac{1}{12}t^3}\) ft/s for the Maserati’s speed for the \(t\!=\!12\) seconds it was recording data as the car passed.

    1. What’s the fastest the Maserati was going in those 12 seconds?
    2. How far did the Maserati travel in those 12 seconds?
    3. What percent of the total distance travelled did the Maserati cover in the first half of that 12 second interval?
    4. What was the Maserati’s average speed over those 12 seconds?
    5. What was the Maserati’s average acceleration over those 12 seconds?
    6. At what \(t\) during those 12 seconds did the Maserati’s acceleration hit a maximum? What was this maximum acceleration?
    7. Suppose that the Maserati passed your location at the exact moment it hit its maximum acceleration. Write down a formula for the function \(F(t)\) that describes the Maserati’s distance from you location at any time \(t\) during that 12 second interval.
  5. State both parts of the Fundamental Theorem of Calculus. Then explain, as if explaining to a high school student taking a calculus class, the significance of the Fundamental Theorem of Calculus.
  6. If \(\rho(t)\) represents the rate at which a population of E. coli grows in a Petri dish \(t\) hours after being introduced, measured in millions of individual bacterial cells per hour, what does the integral \(\int_7^{11} \rho(t) \,\mathrm{d}t\) represent?
  7. If \(\omega(t)\) represents the rate at which water was released from Lake Powell, measured in acre-feet per day, for any day \(t\) of the year 1992, what does the integral \(\int_{92}^{121} \omega(t)\,\mathrm{d}t\) represent?

  8. Recall the first part of the Fundamental Theorem of Calculus, and write down a formula for the derivative of each of the following functions of \(x.\)

    \(\displaystyle a(x) = \int\limits_{-\pi}^{x} t^3\!-\!t \,\mathrm{d}t \)
    \(\displaystyle b(x) = \int\limits_{x}^{7} t^2\sec(t) \,\mathrm{d}t \)
    \(\displaystyle c(x) = \int\limits_{1}^{\tan(x)} \frac{\sin\bigl(\theta\bigl)}{\theta} \,\mathrm{d}\theta \)

  9. None of the following definite integrals can be evaluated; why not?

    \(\displaystyle\int\limits_{-1}^{1} \sqrt{2x+1} \,\mathrm{d}x \)
    \(\displaystyle\int\limits_{1}^{3} \frac{\mathrm{d}x}{(3x-7)^2} \)
    \(\displaystyle\int\limits_{0}^{3} x\sec\bigl(x^2-1\bigr) \,\mathrm{d}x\)

  10. James Stewart The floor function \(\lfloor x \rfloor\) denotes the greatest integer less than \(x\). For example, \(\lfloor 9.81 \rfloor = 9,\) \(\lfloor \pi \rfloor = 3,\) and \(\lfloor -7.3 \rfloor = -8.\) For real numbers \(a\) and \(b\) such that \({0 \lt a \lt b},\) determine an expression for \(\int_a^b \lfloor x \rfloor\,\mathrm{d}x\) in terms of \(a\) and \(b\).

  11. James Stewart Prove that \[ \frac{1}{17} \;\;\leq\;\; \int\limits_1^2 \frac{\mathrm{d}x}{1+x^4} \;\;\leq\;\; \frac{7}{24}\,. \]

  12. James Stewart Prove that if \(f\) is continuous, then \[ \int\limits_0^x f(u)(x-u)\,\mathrm{d}u \;\;=\;\; \int\limits_0^x \left(\int\limits_0^u f(t)\,\mathrm{d}t\right)\,\mathrm{d}u \] How can we interpret this equation geometrically?