5 1 Definite Integral Finding Distance Riemann Sums

5. 1 Definite Integral Finding Distance & Riemann Sums

A car travels 50 m/h for 3 h (3)(50)=150 mile distanc e 150 100 50 D=rt 1 2 3 time

A car travels 50 m/h for 3 hr velocity 150 100 50 (3)(50)=150 mile D=rt v=50 is horizontal line 150 is area time 1 2 3

Ex If V is NOT constant find are Under the curve to find dista

V Ex t v (rate) t D=r t

Ex Left hand Rect Approximate area:

Ex Could also use Right hand Rect Approx area:

To approximate the area un the curve, average the left sum and the right sum. 5. 75+7. 75 = 6. 75 2

Ex Could also use Midpoint Rect Approx area:

In this example there were four subintervals. As the number of subintervals increases, so does the accuracy.

8 subintervals: Interval Width is. 5 Approx area:

Inscribed rectangles are all below the curve: Circumscribed Rectangles are all Above the curve

Pick an integer n. ex n = 10. Now divide the interval into n equal subintervals. a

Endpoints of the new Subinterv a 0, a 1, a 2, . . . , a 10 This is a partition of [a, b]

In each subintervals [ai -1, ai ] pick a number xi and draw a li segment to the x-axis from the point (xi , 0) to a point on the graph of the function, (xi, f(xi)

The area of each rectangle is

The sum of the areas of the rectangles is This is considered a Riemann Sum

If n = 20 If n= 40

As the number of intervals increases, the Riemann Sum converges to a single numbe called the definite Integral.

The integral is an extension of the concept of a sum. The process of finding integrals is called integration. Integration is used to find area under curves

Learn the rules for + and - A

"The sum of wisdom is that time is never lost that is devoted to work. " - Emerson