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Contents
In this lecture we state and prove the Fundamental Theorem of Calculus. This theorem has that name because it relates derivatives and integrals, the two main branches of calculus. We also discuss some resulting techniques for evaluating definite integrals.
Quick Question
What is the area under the graph of f(x) = 1/2 + x/2 and above the interval [0,1]? Now let F(x) = x/2 + x^{2}/4; what is F(1) − F(0)?
Answer
Outline
Outlines for
The Fundamental Theorem of Calculus
Techniques of Integration
Textbook
The Fundamental Theorem of Calculus
Techniques of Integration
Today's Homework
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Quiz
The Fundamental Theorem of Calculus Quiz
Techniques of Integration Quiz
Examples
 The Scholastic Aptitude Test (SAT) is rescaled so that the scores of n people, ranging from 0 to 1600, fit a distribution in the shape of the following function: f(x) = sin(πx/1600) (nπ/3200). What is the probability that a random person will score between 1200 and 1250?
 Find the derivative of a product of a function and an integral.
 Find the derivative of a product of a function and an integral.

Find the integral from 0 to π/4 of tan(x).
 Use integration by parts to find an expression for the integral from 0 to π/4 of sin(x)^{5}.
taking u = (sin(x))^{4} and dv = sin(x) dx.
 Find
the integral of x^{2}e^{x}.
Videos
 Use the fundamental theorem of calculus for definite integrals (1)
Use the fundamental theorem of calculus for definite integrals (2)
 Find the area under y = x^{4} between x = 1 and x = 5
 Find the integral from –1 to 1 of x^{3}
 Doing the chain rule, backwards
 Integrate e^{x}/(1 + e^{2x})
 Integrate (sin(x))^{4} ·(cos(x))^{3}
 Integrate (x + 2)/√(x^{2} + 4x + π)
 Integrate x^{27} + 3sin(x)
 Integrate Ax^{2} + Bx + C
