The Area Question and the Integral Lesson 6.1

Area Under the Curve What does the following demo suggest about how to measure the area under the curve?

Area Under the Curve Using more and more rectangles to approximate the area

The Area Under a Curve Divide the area under the curve on the interval [a,b] into n equal segments Each "rectangle" has height f(xi) Each width is x The area if the i th rectangle is f(xi)•x We sum the areas a b •

Summation Notation We use summation notation Note the basic rules and formulas Summation Formulas, pg 218

Use of Calculator Note again summation capability of calculator Syntax is:  (expression, variable, low, high)

Practice Summation Try these

Limit of a Sum a b For a function f(x), the area under the curve from a to b is where x = (b – a)/n and Consider the region bounded by f(x) = x2 the axes, and the lines x = 2 and x = 3

Limit of a Sum Now So

Limit of a Sum Continuing …

Practice Summation For our general formula: let f(x) = 3 – 2x on [0,1]

The Sum Calculated Consider the function 2x2 – 7x + 5 Use x = 0.1 Let the = left edge of each subinterval Note the sum

The Area Under a Curve The accuracy of the summation will increase if we have more segments As we increase n As n gets infinitely large the summation is exact

The Definite Integral We will use another notation to represent the limit of the summation Upper limit of integration Lower limit of integration The integrand

Example Try Use summation on calculator.

Example Note increased accuracy with smaller x

Limit of the Sum The definite integral is the limit of the sum.

Practice Try this What is the summation? Which gives us Now take limit

Practice Try this one For n = 50? Now take limit What is x? What is the summation? For n = 50? Now take limit

Assignment Lesson 6.1 Page 221 Exercises 1 – 17 odd