Simple Rules for Differentiation



Riemann Sums

Objectives:

Students will be able to

• Calculate the area under a graph using approximation with rectangles.

• Calculate the area under a graph using geometric formulas.

Sometimes it is important to be able to figure out the area of a shape bounded by a curve, the x-axis, and two x-values or between two curves and two x-values. This area can have a tangible meaning. We will concentrate here on the area of a shape bounded a curve, the x –axis and two x-values. If our shape is a basic geometric shape or a combination of two or more basic geometric shapes, we can find the area exactly using geometric formulas.

If the shape is not a basic geometric shape then we will have to approximate the area. The method that we will use will cover the shape with rectangles whose bottom is on the x-axis and whose top is touches the curve at least one point. We then calculate the area of each of the rectangles and add them together to approximate the area under the curve. This method is can Riemann Summation.

As we make the number of rectangles used to approximate the area larger, we get a better approximation. As the number of rectangles approaches infinity, we end up with a definite integral.

For f(x) a continuous function on the interval [pic], the area bounded by the graph of f(x), the x-axis, a, and b using Riemann sums can be represented by

[pic] where [pic] and [pic] is the x-value in the ith subinterval so that [pic] touches the graph. As n approaches infinity, this can be represented as the definite integral [pic]

In general it is easier for the width of each rectangle to be equal. As for the value of x that we use, there are some standard values that are often chosen. We can use the left side of each subinterval, the right side of each subinterval, or the midpoint of each subinterval.

Example 1:

Find [pic] for the graph of f(x) shown below

[pic]

Example 2:

Approximate the area under the graph of [pic] and above the x-axis from x = 1 to x = 9 using rectangles with n = 4 for each of the following methods:

a. left endpoints

b. right endpoints

c. average the answers to parts a and b

d. midpoints

Example 3:

Find the exact value of the integral [pic] using formulas from geometry.

Example 4:

Find the exact value of the integral [pic] using formulas from geometry.

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