- Find the total area boundedby the graphof and the -axis,from to to
- Use your knowledgeof geometryto evaluatethe followingdefiniteintegral:
(Hint:set and squareboth sidesto see if you can recognizethe region
from geometry.)
Answers
a. b. c. d.
a. b. c. d.
The graphis symmetricaboutthe origin;hence
The graphis that of a quartercircleof radius ; hence
EvaluatingDefiniteIntegrals
LearningObjectives
- Use antiderivativesto evaluatedefiniteintegrals
- Use the MeanValue Theoremfor integralsto solveproblems
- Use generalrulesof integralsto solveproblems
Introduction
In the Lessonon DefiniteIntegrals,we evaluateddefiniteintegralsusingthe limit definition.This process
was long and tedious.In this lessonwe will learnsomepracticalwaysto evaluatedefiniteintegrals.We
beginwith a theoremthat providesan easiermethodfor evaluatingdefiniteintegrals.Newtondiscovered
this methodthat usesantiderivativesto calculatedefiniteintegrals.
Theorem: