Worksheet for Section 5.2

Secton 5.2 is about area, and ways to compute area of regions in the plane using approximations. To make the approximations easier to write out, use the summation notation:

$$\sum_{i=1}^n a_i=a_1+a_2+a_3+\dots+a_n$$

Here are some useful formulas written using the summation notation:

$$\sum_{i=1}^n c=cn\qquad\qquad \sum_{i=1}^n i=\frac{n(n+1)}2$$

$$\sum_{i=1}^n i^2=\frac{n(n+1)(2n+1)}6\qquad\qquad$$

As an example, use these formulas to compute the sum $$\sum_{i=1}^{10} (i+1)(i+3)$$.

This summation notation can be used to help approximate the area of a plane region, such as the region between graph of $f(x)=4-x^2$ and the $x$-axis, between $x=0$ and $x=2$, by adding up areas of easier regions. For example, here is an approximation using four rectangles:

   Area$$\approx \sum_{i=1}^4 f\left(\frac{i}2\right)\left(\frac12\right) =\sum_{i=1}^4 \left(4-\left(\frac{i}2\right)^2\right)\left(\frac12\right)$$

This approximation uses rectangles that are all under the graph. How can you modify the sum to find an approximation using rectangles that are all over the graph? What do these two approximations tell you about the actual area of the region?

A Riemann sum for $f(x)$ on the interval $[a,b]$ is a sum of the form $$\sum_{i=1}^n f(c_i)\Delta t_i$$, where $a=t_0<t_1<\dots<t_n=b$, and for $i=1,2,\dots,n$, $\Delta t_i=t_i-t_{i-1}$ and $t_{i-1}\leq c_i\leq t_i$. In the examples we have been looking at, all of the $\Delta t_i$ take the same value $\Delta t$ (equal-length subintervals), but that is not necessary. A right-hand sum would use $c_i=t_i$; a left-hand sum would use $c_i=t_{i-1}$. The Riemann Sums applet on the course web page lets you compute these Riemann sums (with equal-length subintervals) for left-hand sums, right-hand sums, midpoints, or other points in the subintervals. The definite integral is the limit of the left-hand or right-hand sum as $n$ approaches $\infty$ -- this can be written as follows:

$$\int_a^b f(x)\, dx=\lim_{n\to\infty}\sum_{i=1}^n f(t_i)\Delta t$$

Note that the sum above is written as a right-hand sum, with $c_i=t_i$. You can do the same with the more general Riemann sums, but if the subintervals are not equal length, then it is a bit more complicated than just using a limit as $n\to\infty$ -- you have to be sure that the length of the widest subinterval approaches 0.

If $f(x)\geq 0$ on the interval $[a,b]$ (where $a<b$), then the definite integral corresponds to the area under the graph of $f(x)$ and above the $x$-axis, between $a$ and $b$. For example, the definite integral $$\int_{-1}^1\sqrt{1-x^2}\, dx$$ can be computed using Riemann sums, but the area under this graph also has a simple geometric shape, and that can be used to find the exact value of the definite integral. (What is the shape of the region?) Explore the Riemann sums using the applet, and compare the values of the approximations with the exact value obtained geometrically.

For functions which take negative values in the interval, recall that the terms in the Riemann sums corresponding to rectangles under the $x$-axis are actually taken to have negative area, so that areas under the $x$-axis can cancel areas over the $x$-axis. Consider, for example, the definite integral $$\int_0^2 x^2-1\, dx$$. Graph the area and consider approximations by Riemann sums, again using the applet. We will begin to see in the next section how these area computations relate to derivatives, which will give better ways to find the exact values of definite integrals.