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]. Examples 9. Find the area of the region bounded by the curve [pic]and the x-axis. 10. Find the area between the x-axis and the curve [pic] from[pic]. Answer to examples 1. 1642. [pic] 3. [pic] 4. [pic] 5. [pic] 6. [pic] 7. [pic] 8. [pic] 9. [pic] 10. [pic]