Area and the Definite Integral Lesson 7.3A

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 •

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 Where 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 7.3A Page 458 Exercises 6 – 20 all