The function is a function of single variable . is the dependent variable and is the independent variable. The indefinite integral
is a function, such that
The definite integral
gives the area under the curve , bounded by the lines , and the axis.
These are examples of single integrals, where only 1 independent variable is present. If the number of independent variables in a function is greater than 1, multiple integrals are used.
For functions with 2 independent variables, , double integrals are evaluated and for functions with 3 independent variables, , triple integrals are evaluated. It will soon become clear that
I) Single integrals are evaluated over a line.
II) Double integrals are evaluated over an area and
III) Triple integrals are evaluated over a 3D space.
Let be a function of 2 independent variables and , defined over a region , bounded by a closed curve . Let this region be divided into smaller regions, such that th region has area . Let be a point inside the th region. Then the double integral over the region is defined as
A critical part in evaluating multiple integrals is choosing the upper and lower limits. For double integrals, the limits of the inner integral are to be expressed as functions of the other variable, which corresponds to the outer integral. For example, consider the figure below:
Thus, the inner integral is to be evaluated w.r.t. , considering as a constant. Having obtained the inner integral, now, we should evaluate the outer integral as a single integral (the usual one). The limits of outer integral will be constants ( and in this case).
If the limits of the inner integrals are functions of , the outer integral will be w.r.t and the inner integral must be evaluated w.r.t. .
Note that order of integration DOES NOT matter, as long as the limits are appropriately put.
I) Sometimes, it is required to change the order of integration, when the inner integral is impossible to integrate. In such cases, the limits are obtained by plotting the graph.
II) If the integrand involves terms of the form or , it becomes easier, if we transform the integrand to polar coordinates,
Generally, the inner integral is obtained w.r.t. .
III) If , we get the area of the region.
Evaluate the following over the region .
The region of integration is plotted below :
Clearly, to the left of line , the region is absent, hence the lower limit on will be . The upper limit will be . No restriction on the highest value of .
Let’s now consider a strip parallel to axis. It will first intersect the region at . So, the lower limit on is . The upper limit will be because no curve bounds the region in that direction. Hence, given integral will be
Since inner limits are of , let’s first integrate w.r.t. .
As stated earlier, a triple integral is evaluated over a . Let there be a function defined over a closed region having volume . Let the region be divided into subregions, such that the th subregion has volume . Let be a point in th subregion. Then the triple integral is given by
The limits of integrals are equally important. The innermost integral (say w.r.t. ) has limits expressed in terms of functions of the other 2 variables, . Once it is evaluated treating and as constants, it simplifies to a double integral problem and can be solved as explained earlier.
I) Transforming the integral into spherical polar coordinates requires the following replacement:
II) Transforming the integral into cylindrical polar coordinates requires the following replacement:
III) For a sphere , the limits are to , to , to
IV) For a hemisphere , the limits are to , to , to
IV) Dirichlet’s Theorem for triple integrals :
If , then
Areas and Volumes
I) When in a double integral, the function for all values of in the region, the double integral gives the area of the region.
II) When in a triple integral, the function for all in the closed region, the triple integral gives the volume of the region.
III) Other applications include obtaining center of mass, center of gravity, mean values, RMS values, moment of inertia etc.