AREA Section 4.2

When you are done with your homework, you should be able to… Use sigma notation to write and evaluate a sum Understand the concept of area Approximate the area of a plane region Find the are of a plane region using limits

SIGMA NOTATION The sum of n terms is written as where i is the index of summation, is the ith term of the sum, and the upper and lower bounds of summation are n and 1. *The lower bound can be any number less than or equal to the upper bound

Evaluate

Evaluate

Summation Properties 1. 2.

Theorem: Summation Formulas 1. 2. 3. 4.

AREA Recall that the definition of the area of a rectangle is From this definition, we can develop formulas for many other plane regions

AREA continued… We can approximate the area of f by summing up the areas of the rectangles: What happens to the area approximation when the width of the rectangles decreases?

Theorem: Limits of the Lower and Upper Sums Let f be continuous and nonnegative on the interval . The limits as of both the lower and upper sums exist and are equal to each other. That is, Where and and are the minimum and maximum values of f on the subinterval.

Continued… Since the same limit is attained for both the minimum value and the maximum, it follows from Squeeze Theorem that the choice of x in the ith subinterval does not affect the limit. This means that we can choose an arbitrary x-value in the ith subinterval.

Definition of an Area in the Plane Let f be continuous and nonnegative on the interval . The area of the region bounded by the graph of f, the x-axis, and the vertical lines and is

Find the area of the region bounded by the graph , the x-axis, and the vertical lines and .