1. Does: ? 2. What is: ? Think about:

Finding Area between a Function & the x-axis Chapters 5.1 & 5.2 January 25, 2007

Using Data Underestimate the distance traveled: t: v:

Ex: y=x 2 on [1,3] L 4 Length of subintervals (rectangles) = ∆x = (3 - 1)/4 =.5 Data points: x: y: f(1) f(1.5) f(2) f(2.5) f(3) 1 9/4 4 25/4 9

L4L4

R4R4

T4T4

Trapezoid Rule: T n = (L n +R n )/2 OR T n = (∆x/2)[f(x 0 ) + 2f(x 1 ) + 2f(x 2 ) + … + 2f(x n-1 ) + f(x n )] Where n = # of sub-intervals (rectangles) ∆x = (b - a)/n

M4M4

Midpoint Rule: M n = ∆x[f(x 0 + ∆x/2) + f(x 1 + ∆x/2) + f(x 2 + ∆x/2) + … + f(x n-1 + ∆x/2) ] Where n = # of sub-intervals (rectangles) ∆x = (b - a)/n

You can use these approaches to find the area of “odd shapes”

L 10

R 10

T 10

Estimate the area using M 3 : F(x)=3/(1+x 2 ) on the interval [-1,5] Applet

Definition The area of A of the region S that lies under the graph of the continuous function f is the limit of the sum of the areas of approximating rectangles: where:

Use the Def n to find an expression for the exact area under: y=x 2 on the interval [1,3] f(x) = 3/(1+x 2 ) on the interval [-1, 5] g(x) = ln[x]/x on the interval [3,10]

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:

Use the Def n to find an expression for the exact area under: y=x 2 on the interval [1,3] f(x) = 3/(1+x 2 ) on the interval [-1, 5] g(x) = ln[x]/x on the interval [3,10]

Express the limit as a Definite Integral

Express the Definite Integral as a limit

Building on the idea of Area to evaluate the Definite Integral where c is a constant. Look at the graph…….

Building on the idea of Area to evaluate the Definite Integral Again, look at the graph…….

Building on the idea of Area to evaluate the Definite Integral Again, look at the graph…….

Building on the idea of Area to evaluate the Definite Integral Again, look at the graph…….

Building on the idea of Area to evaluate the Definite Integral Again, look at the graph…….

Properties of the Definite Integral