The Definite Integral and Area Under a Curve

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  • Topic: Integral, Derivative, Fundamental theorem of calculus
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Lecture 15 The Definite Integral and Area Under a Curve Definite Integral ---The Fundamental Theorem of Calculus (FTC)

Given that the function [pic] is continuous on the interval [pic] Then, [pic]
where F could be any antiderivative of f on a ( x ( b. In other words, the definite integral [pic] is the total net change of the antiderivative F over the interval from [pic]

• Properties of Definite Integrals (all of these follow from the FTC) 1. [pic]4. [pic]
2. [pic]5. [pic]
3. [pic], k is a constant.

Examples
1. Find [pic]2. Find [pic]

3.Suppose [pic]. Find [pic], hence find [pic]
4.Suppose [pic]. Find.[pic].

• Evaluate Definite Integrals by Substitution

The method of substitution and the method of integration by parts can also be used to evaluate a definite integral.

[pic]
Examples
5. Find [pic] 6.Find [pic]
7.Find [pic]8.Find [pic]
Area and Integration

There is a connection between definite integrals and the geometric concept of area. If f(x) is continuous and nonnegative on the interval [pic], then the region A under the graph of f between [pic]has area equal to the definite integral [pic]. [pic],

where [pic]is any antiderivative of [pic].

• Why the Integral Formula for Area Works?

[pic]

Let A(x) denote the area of the region under f between a and x, then

[pic]

In general,

[pic]
[pic]
By the definition of the derivative,

[pic]
[pic]
[pic]
Since A(a) is the area under the curve between x = a and x = a, which is clearly zero. Hence, [pic]= the area under f between a and b.

Note:The fundamental theorem requires that the function [pic] is non-negative over the interval [pic]. If [pic]is negative over the interval [pic], the definite integral, [pic], results in a value that is the negative of the area between [pic]and the x-axis from[pic] In such a case, the area between the x-axis and the curve is the absolute value of the definite integral, [pic]....
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