Worksheet for Section 6.4

Secton 6.4 is about the Second Fundamental Theorem of Calculus. The Fundamental Theorem of Calculus from Chapter 5 can be interpreted as establishing an inverse relationship between derivatives and definite integrals. To extend this idea, the Second Fundamental Theorem looks at what happens if you reverse the operations -- take a derivative of a definite integral. In order to do this, you need to be able to treat a definite integral as a function, by allowing for a variable in one of the endpoints. For a function $f(x)$, define $F(x)$ as follows:

$$F(x)=\int_a^x f(x)\, dx$$

If $F(x)$ is defined this way, what is $F'(x)$? Compute $F'(x)$ using the limit definition of the derivative and the comparison theorems for definite integrals from Chapter 5. (I will do this in class.) The result is that $F'(x)=f(x)$ -- in other words, $F(x)$ is an antiderivative for $f(x)$. This can be thought of as a technique for constructing an antiderivative for $f(x)$. (Using the fact that, as defined above, $F(a)=0$, the antiderivative $F(x)$ constructed this way can even be used to prove the original Fundamental Theorem.) Here is another way of writing Second Fundamental Theorem:

$$\frac{d}{dx}\int_a^x f(t)\, dt=f(x)$$

For example, if $$F(x)=\int_1^x \frac1t\, dt$$, find $F'(x)$; as a more complicated example, if $$F(x)=\int_0^{x^2} \sin t\, dt$$ -- again, find $F'(x)$.

A more interesting example that appears in some applications is the sine-integral function:

$$Si(x)=\int_0^x \frac{\sin x}{x}\, dx$$

Using the Riemann Sums applet, estimate values for $Si(1)$, $Si(1.5)$, and $Si(2)$. What is the derivative of $Si(x)$? What is the derivative of $g(x)=x\cdot Si(x)$?