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