Understanding Double Integrals in Mathematics
Learn about double integrals, their application in calculating volumes of three-dimensional solids, and how they are used to evaluate regions in the xy-plane. Explore exercises and definitions related to double integrals, along with Fubini's theorem and the concept of iterated integrals.
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Lecture Ten Lecture Ten Double Integrals Double Integrals
Integrals of a function of two variables over a region in R2are called double integrals. Just as the definite integral of a positive function of one variable represents the area of the region between the graph of the function and the x-axis, the double integral of a positive function of two variables represents the volume of the region between the surface defined by the function and the plane which contains its domain.
A double integral is an integral used to evaluate the volume of a three dimensional solid under the surface z = f(x,y) with respect to x and y. A double integral is an iterated integral. The integral is iterated twice.
Exercises: Q1. Evaluate the following double integrals: Exercises: Q2. Show that:
Definition the double integral: Definition: The AREA A AREA A of a region R in the x y- plane is given by Definition and above a region R in the x y- plane is: Definition: The VOLUME V VOLUME V beneath the surface Fubini' If f (x, y) is continuous on R=[a,b] X [c,d]. then, Fubini's s Theorem: Theorem: These integrals are called iterated integrals.
Note and are fixed , meaning the ranges of x and y don't depend on each other. Note: Rectangular regions are easy because the limits For regions of other shapes, the range of one variable will depend on the other. In a double integral, but the inner limits can depend on the outer variable In a double integral, the outer limits must be constant, but the inner limits can depend on the outer variable. the outer limits must be constant,
