In this section, we will investigate how to estimate the value of a definite integral when geometry fails us. We will also construct the formal definition of the definite integral.

Section 5.6 Approximating Sums

Formal Definition of the Definite Integral

Idea

Consider the function f(x) = sin(x) + 2 and its integral over the interval [1, 3].

Idea

Consider the function f(x) = sin(x) + 2 and its integral over the interval [1, 3].

The area of this region is not able to be found using basic geometric formulas like we did in section 5.1

Idea

Consider the function f(x) = sin(x) + 2 and its integral over the interval [1, 3].

The area of this region is not able to be found using basic geometric formulas like we did in section 5.1

Instead, we estimate the value of the area by using some number of rectangles (left, right, or midpoint) or trapezoids.

Rectangle Process

1. Divide the interval [a, b] into n subintervals each of width

2. For Ln or Rn, use the appropriate point from each subinterval (called xi). The height of each rectangle is and its area is .

3. For Mn, use the midpoint of each subinterval (called mi). The height of each rectangle is and its area is .

4. Add together the area of the n rectangles to get the desired approximation.

Trapezoid Process

1. Divide the interval [a, b] into n subintervals each of width

2. In the ith subinterval, use both of the endpoints of that subinterval, xi and xi+1. The area of this trapezoid is .

3. Add together the area of the n trapezoids to get the desired approximation.

4 Left Rectangles

Consider the function f(x) = sin(x) + 2 and its integral over the interval [1, 3].

We use the endpoints:

4 Right Rectangles

Consider the function f(x) = sin(x) + 2 and its integral over the interval [1, 3].

We use the endpoints:

4 Midpoint Rectangles

Consider the function f(x) = sin(x) + 2 and its integral over the interval [1, 3].

We use the midpoints:

4 Trapezoids

Consider the function f(x) = sin(x) + 2 and its integral over the interval [1, 3].

We use all the endpoints:

Example 1

Find the L4, R4, M4, and T4 approximation for

Question?

What can we do to get more accurate approximations?

Question?

What can we do to get more accurate approximations?

Use midpoint rectangles.

They are, in general, more accurate than left or right rectangles or

trapezoids.

Question?

What can we do to get more accurate approximations?

Use midpoint rectangles.

They are, in general, more accurate than left or right rectangles or

trapezoids.

Use a larger value of n The more shapes used, the more accurate the approximation

Definition

The definite integral of f from x = a to x = b is formally defined by: