Fundamental Theorem Part 2 Discussion

(The following is/are the initial prompt(s) for an online discussion; students may have posted responses, and prompts for further discussion may have been added, but these things are not shown.)

Part 2 of the Fundamental Theorem of calculus gives you a way to evaluate definite integrals in terms of antiderivatives. For that reason it’s also sometimes called the “evaluation theorem.” Our textbook talks about this theorem in section 5.3, under “The Fundamental Theorem of Calculus, Part 2: The Evaluation Theorem.” The discussion below gives you a chance to start using it.

Evaluation

The most straightforward use of part 2 of the Fundamental Theorem is to evaluate definite integrals. For example, in our November 18 class, we argued on symmetry and area grounds that

Can you use the Fundamental Theorem to show that this is in fact right?

Can you use the Fundamental Theorem to evaluate the following integrals?

1. \(\int_0^{\pi} \sin x\,dx\)
2. \(\int_1^3 2x^2 - x\,dx\)
3. \(\int_1^4 \sqrt{x}\,dx\)

Applications

Now that we have an easy way to evaluate integrals, it becomes more practical to start looking at some applications of them. For example...

Can you work out what the average value of the function \(f(x) = x^2\) is over the interval \([-1,1]\)?

Suppose a car accelerates in such a manner that its speed after \(t\) seconds is given by \(s(t) = 3t + 2\). Can you figure out how far it travels in the first 5 seconds?