Unit 7A

Integrals Part 1

Lesson 4: Areas Part 2

In the previous Lesson, the Fundamental Theorem of Calculus was introduced, and a more efficient way of evaluating the following limit was developed.


In fact, the sum was called the definite integral of function «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/math» from «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»a«/mi»«/math» to «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»b«/mi»«/math». And, the definite integral was represented by «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msubsup»«mo»§#8747;«/mo»«mi»a«/mi»«mi»b«/mi»«/msubsup»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mi»d«/mi»«mi»x«/mi»«/math».

Recall this integral can be evaluated by finding the antiderivative «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»F«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/math», evaluating «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»F«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/math» at «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«mo»=«/mo»«mi»a«/mi»«/math» and «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«mo»=«/mo»«mi»b«/mi»«/math», and then finding the difference between the two, «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»F«/mi»«mfenced»«mi»b«/mi»«/mfenced»«mo»§#8722;«/mo»«mi»F«/mi»«mfenced»«mi»a«/mi»«/mfenced»«/math».

Fundamental Theorem of Calculus If «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»y«/mi»«mo»=«/mo»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/math» is continuous on the interval «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfenced open=¨[¨ close=¨]¨»«mrow»«mi»a«/mi»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mi»b«/mi»«/mrow»«/mfenced»«/math», then «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msubsup»«mo»§#8747;«/mo»«mi»a«/mi»«mi»b«/mi»«/msubsup»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mi»d«/mi»«mi»x«/mi»«mo»=«/mo»«mi»F«/mi»«mfenced»«mi»b«/mi»«/mfenced»«mo»-«/mo»«mi»F«/mi»«mfenced»«mi»a«/mi»«/mfenced»«/math», where «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»F«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/math» is an antiderivative of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/math».

If «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»y«/mi»«mo»=«/mo»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/math» is a non-negative continuous function on the interval «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfenced open=¨[¨ close=¨]¨»«mrow»«mi»a«/mi»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mi»b«/mi»«/mrow»«/mfenced»«/math», one interpretation of the definite integral is the area bounded by the graph of the function, the «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«/math»-axis, and the vertical lines «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«mo»=«/mo»«mi»a«/mi»«/math» and «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«mo»=«/mo»«mi»b«/mi»«/math».

Because the definite integral is based on the indefinite integral «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»F«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/math», the properties of the indefinite integral can be extended to the definite integral. As such, the following properties apply.

Let «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/math» and «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»g«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/math» be continuous functions defined on the interval «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfenced open=¨[¨ close=¨]¨»«mrow»«mi»a«/mi»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mi»b«/mi»«/mrow»«/mfenced»«/math» and let «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»c«/mi»«/math» be a real constant.


The third property states that switching the limits of integration changes the sign of the integral.