Volume using Double Integral

Volume using Double Integral is the geometric interpretation of the double integral, to calculate the volume using double integral, let’s consider a region R over [a × b] and . A curve S = f(x, y) is drawn such that it projects an area in this region R. The graph for this is shown below:

In order to calculate the volume of the shown region, we will divide the length cd of the area R into m equal parts and the breadth of the area R into n equal parts. Thus the complete area R gets divided into smaller rectangles. Now from each of these rectangles we will draw a box upwards to the point where it meets S. It can be shown as follows:

Consider that the area of the base of each small rectangle is A and its height is given by f(x’, y’). The volume under region S can be calculated using:

[Tex]\bold{V = \Sigma_{i=1}^n \Sigma_{j=1}^n A.f(x’,y’)} [/Tex]

We have made use of two variables m and n because the volume will be added and calculated in both directions or axes i.e. X axis and Y axis. As the value of m and n becomes very large and approaches infinity which means the area R is divided into infinite small rectangular regions, then this equation can be written as:

[Tex]V=\lim_{m,n \rarr\infty}\sum_{i=1}^m\sum_{j=1}^nA.f(x’,y’) [/Tex]

We know that when the variables tend to infinity then summation can be substituted with integration which is also the definition of integration. Thus above equation becomes equal to:

[Tex]V= \int\int_Rf(x,y)dA [/Tex]

Thus we derive the formula to find the volume under the curve using a double integral.

A double integral is a mathematical tool for computing the integral of a function of two variables across a two-dimensional region on the xy plane. It expands the concept of a single integral by integrating the functions of two variables over regions, surfaces, or areas in the plane. In case two variables are present, we need to substitute the value of one variable in terms of the other. This technique becomes very difficult when we deal with multiple variables to calculate the areas and volumes under the curves. A double integral is very useful in such cases. In this article, we will learn about double integrals in detail.