Homework questions thus far??? Section 4.10? 5.1? 5.2?

The Definite Integral Chapters 7.7, 5.2 & 5.3 January 30, 2007

Estimating Area vs Exact Area

Pictures Riemann sum rectangles, ∆t = 4 and n = 1:

Better Approximations Trapezoid Rule uses straight lines

Trapezoidal Rule

Better Approximations The Trapezoid Rule uses small lines Next highest degree would be parabolas…

Simpson’s Rule Mmmm… parabolas… Put a parabola across each pair of subintervals:

Simpson’s Rule Mmmm… parabolas… Put a parabola across each pair of subintervals: So n must be even!

Simpson’s Rule Formula Like trapezoidal rule

Simpson’s Rule Formula Divide by 3 instead of 2

Simpson’s Rule Formula Interior coefficients alternate: 4,2,4,2,…,4

Simpson’s Rule Formula Second from start and end are both 4

Simpson’s Rule Uses Parabolas to fit the curve Where n is even and ∆x = (b - a)/n S 2n =(T n + 2M n )/3

Use Simpson’s Rule to Approximate the definite integral with n = 4 g(x) = ln[x]/x on the interval [3,11] Use T 4.

Runners: A radar gun was used to record the speed of a runner during the first 5 seconds of a race (see table) Use Simpsons rule to estimate the distance the runner covered during those 5 seconds.

Definition of Definite Integral: If f is a continuous function defined for a≤x≤b, we divide the interval [a,b] into n subintervals of equal width ∆x=(b-a)/n. We let x 0 (=a),x 1,x 2,…,x n (=b) be the endpoints of these subintervals and we let x 1 *, x 2 *, … x n * be any sample points in these subintervals so x i * lies in the ith subinterval [x i-1,x i ]. Then the Definite Integral of f from a to b is:

Express the limit as a Definite Integral

Express the Definite Integral as a limit

Properties of the Definite Integral

Properties of the Integral 1) 2) = 0 3) for “c” a constant

Properties of the Definite Integral Given that:  Evaluate the following:

Properties of the Definite Integral Given that:  Evaluate the following:

Given the graph of f, find:

Evaluate:

Integral Defined Functions Let f be continuous. Pick a constant a. Define:

Integral Defined Functions Let f be continuous. Pick a constant a. Define: Notes: lower limit a is a constant.

Integral Defined Functions Let f be continuous. Pick a constant a. Define: Notes: lower limit a is a constant. Variable is x: describes how far to integrate.

Integral Defined Functions Let f be continuous. Pick a constant a. Define: Notes: lower limit a is a constant. Variable is x: describes how far to integrate. t is called a dummy variable; it’s a placeholder

Integral Defined Functions Let f be continuous. Pick a constant a. Define: Notes: lower limit a is a constant. Variable is x: describes how far to integrate. t is called a dummy variable; it’s a placeholder F describes how much area is under the curve up to x.

Example Let. Let a = 1, and. Estimate F(2) and F(3).

Example Let. Let a = 1, and. Estimate F(2) and F(3).

Where is increasing and decreasing? is given by the graph below: F is increasing. (adding area) F is decreasing. (Subtracting area)

Fundamental Theorem I Derivatives of integrals: Fundamental Theorem of Calculus, Version I: If f is continuous on an interval, and a a number on that interval, then the function F(x) defined by has derivative f(x); that is, F'(x) = f(x).

Example Suppose we define.

Example Suppose we define. Then F'(x) = cos(x 2 ).

Example Suppose we define. Then F'(x) =

Example Suppose we define. Then F'(x) = x 2 + 2x + 1.

Examples:

If f is continuous on [a, b], then the function defined by is continuous on [a, b] and differentiable on (a, b) and Fundamental Theorem of Calculus (Part 1)

If f is continuous on [a, b], then the function defined by is continuous on [a, b] and differentiable on (a, b) and Fundamental Theorem of Calculus (Part 1) (Chain Rule)

In-class Assignment a. Estimate (by counting the squares) the total area between f(x) and the x- axis. b. Using the given graph, estimate c. Why are your answers in parts (a) and (b) different? Find:

First let the bottom bound = 1, if x >1, we calculate the area using the formula for trapezoids: Consider the function f(x) = x+1 on the interval [0,3]

Now calculate with bottom bound = 1, and x < 1, : Consider the function f(x) = x+1 on the interval [0,3]

So, on [0,3], we have that And F’(x) = x + 1 = f(x) as the theorem claimed! Very Powerful! Every continuous function is the derivative of some other function! Namely: