Worksheet for Section 5.3

Secton 5.3 is about the Fundamental Theorem of Calculus, the connection (finally!) between area and derivatives. If $f(x)=F'(x)$, then the definite integral of $f(x)$ over $[a,b]$ can be computed using $F(x)$ as follows:

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

Needless to say, this saves a lot of work, if you can find $F(x)$ given $f(x)$. More on this later -- in each of the examples below, the function $F(x)$ should be pretty easy to find from the given $f(x)$ in the definite integral. In each case, compute the definite integral using the Fundamental Theorem:

$$\int_0^2 x^2\, dx\qquad \int_0^4 \sqrt{x}\, dx\qquad \int_{-1}^2 x\, dx$$

$$\int_0^1 e^x\, dx\qquad \int_0^\pi \sin x\, dx\qquad \int_0^e \frac1x\, dx$$

This section emphasizes the interpretation of the Fundamental Theorem. The relationship $f(x)=F'(x)$ indicates that $f(x)$ gives the rate of change in $F(x)$ -- in terms of position and velocity, $F(t)$ is the position function and $f(t)$ is the velocity function (written now in terms of time $t$). The definite integral gives the total change in $F(t)$ between $t=a$ and $t=b$. To calculate the total change in position, subdivide the interval $[a,b]$ into $n$ many subintervals.... In other words, compute a Riemann sum to determine the definite integral of the rate of change $f(t)$ over the interval $[a,b]$. On the other hand, if you know the position function $F(t)$, then the total change in position between $t=a$ and $t=b$ is just $F(b)-F(a)$. In words: the definite integral of the rate of change is equal to the total change. I will look at some application problems in class related to this interpretation.

The average value of a function $f(x)$ on an interval $[a,b]$ is given by:

   Average value$$=\frac1{b-a}\int_a^b f(x)\, dx$$

For example, find the average value of $f(x)=2x^2+1$ on the interval $[-1,3]$. Graph $f(x)$ on $[-1,3]$, and draw a horizontal line corresponding to your computed average value -- what does the graph indicate about the average value of $f(x)$? Now find the average value of $\sin x$ on $[0,\pi]$, and again graph the function and a horizontal line at the average value.