The Definite Integral and Area Under a Curve: The Fundamental Theorem of Calculus
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,
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,