Math 5A: Area Under a Curve

The Problem: Find the area of the region below the curve f(x) = x 2 +1 over the interval [0, 2].

IDEAS?

IDEA Approximate Area by: Area Triangle + Area Rectangle = (2)(1) + (1/2)(2)(4) = 6 Actual Area _______ 6

IDEA: Vertical strips Cut the region into vertical strips. Cut the top horizontally to make rectangles. Approximate the area under the curve using the rectangles.

4 rectangles Each rectangle Width= = =1/2 Height determined by functional value. Area of rectangles= Actual Area ____3.75

4 rectangles-right endpoint Each rectangle Width= = =1/2 Height determined by functional value at rt endpt. Area of rectangles= Actual Area ____5.75

Caution Left endpoint does not always yield an underestimate, nor right an overestimate. Consider f(x) = sinx on [0,  ]

4 rectangles-midpoint Each rectangle Width= = =1/2 Height determined by functional value at midpoint. Area of rectangles= Actual Area ____4.625

Improving Area Estimate Left Endpoint - 8 Rectangles Width = = =1/4 Area of Rectangles

More Rectangles-Left Endpoint # rect= Area=

More Rectangles-Right Endpoint # rect= Area=

Estimate Appears to Approach … Regardless of Point Chosen Left Endpoint Right Endpoint Midpoint

Choose Random Point

Best Estimate As the number of rectangles, n, increases, the area in the rectangles appears to approach the area under the curve, regardless of point chosen. Define Area Under the Curve =

Generalizing… f(x) conts. and f(x) 0 on [a,b] Partition [a,b] a random point in

Formalize Example If choose right endpoint

Example - continued BIG STEP HERE !! What happened?

Example continued So…AREA = Corresponds to earlier approximation of ….

Things to Think About What if f(x) <0 on all or part of [a,b]? Can we always find a closed form expression for ? What do we do when we can’t? Remember the big step?