How to Calculate Double Integral?
Steps to calculate the double integral are as follows:
Step 1: Write down the function to be integrated with a double integral sign and mention the upper and lower limits of integration on the integral.
Step 2: Integrate the function with respect to any one of the variables initially.
Step 3: Now insert the lower and upper limits of the variable with respect to which we integrated the function in order to bring the left function into one variable.
Step 4: Proceed similarly as above, integrate the function again with respect to the other variable.
Step 5: Insert the lower and upper limits of the second variable to get the result.
Example: Consider the integration [Tex]\bold{\int^2_1\int^2_1(x+y)dy~dx} [/Tex].
Solution:
In order to solve this integral, we will solve the inner integral first and integrate the function w.r.t x
Let [Tex]I = \int^2_1\int^2_1(x+y)dy~dx [/Tex]
[Tex]\Rightarrow I = \int^2_1[\frac{x^2}{2}+xy]^{x=2}_{x=1} [/Tex]
Putting the upper and lower limits for x, we get
⇒ I [Tex]= \int^2_1[\frac{2^2}{2}+2y-(\frac{1^2}{2}+y)]dy [/Tex]
⇒ I [Tex]= \int^2_1[2+2y-\frac{1}{2}-y]dy [/Tex]
⇒ I [Tex]= \int^2_1(\frac{3}{2}+y)dy [/Tex]
Now integrating w.r.t y, we get
⇒ I [Tex]= [\frac{3y}{2}+\frac{y^2}{2}]^{y=2}_{y=1} [/Tex]
⇒ I [Tex]= [\frac{3(2)}{2}+\frac{2^2}{2}-(\frac{3(1)}{2}+\frac{1^2}{2})] [/Tex]
⇒ I = 3 + 2 – 2 = 3
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.