Definite Integral
Example: (Function in one variable)
(Function in one variable)
Partial Integration
• The integral on the right side is called double integral.
(Function in 2 variables)
Consider,
• It is also called iterated integral.
• The double integral consists of an inner (first evaluated) and an outer integrals.
Double Integral
• Usually, the brackets are omitted.
• Means that:
• Thus,
Double Integral
• Similarly, the iterated integral
• Means that:
Example
• For example,
• The same is obtained if we interchange the order of integration
Note: When region R is rectangular, just swap the integrals (do remember to swap dx and dy as well)
Fubini’s Theorem
• Visualization
o Fubini’s Theorem
Thus, a double integral can be evaluated by iterated integrals and the order of integration can be changed, only if each of a, b, c, d is a number, not function.
Given that,
Double Integrals over Nonrectangular Regions
• Iterated integrals with nonconstant limits of integration
Given that,
Triple Integral
• Just as we define single integrals for functions of one variable and double integrals for functions of two variables, so we can define triple integrals for functions of three variables.
• To evaluate triple integrals , we need to express them as iterated integrals, as follows.
• Means that:
Given that,
Given that,
Given that,
