Solved Examples of Double Integral
Example 1: Calculate [Tex]\bold{\int^2_1\int^4_3 xy~dy~dx} [/Tex].
Solution:
Let [Tex]I = \int^2_1\int^4_3xy~dy~dx [/Tex]
Putting the upper and lower limits for x, we get
[Tex]\Rightarrow I = \int^2_1[\frac{x^2y}{2}]^{x=4}_{x=3}dy\\ \Rightarrow I = \int^2_1[\frac{4^2y}{2}-\frac{3^2y}{2}]dy [/Tex]
[Tex]\Rightarrow I = \int^2_1[\frac{x^2y}{2}]^{x=4}_{x=3}dy = \int^2_1\frac{7y}{2}dy [/Tex]
Now integrating w.r.t y, we get
[Tex]\Rightarrow I = \int^2_1[\frac{x^2y}{2}]^{x=4}_{x=3}dy = [\frac{7y^2}{4}]^{y=2}_{y=1} [/Tex]
[Tex]\Rightarrow I = \int^2_1[\frac{x^2y}{2}]^{x=4}_{x=3}dy = [\frac{7(2^2)}{4}-\frac{7(1^2)}{4}] [/Tex]
⇒ I = 7 – 7/4 = 21/4
Example 2: Calculate [Tex]\bold{\int^2_1\int^4_3 (2x + 3y)~dy~dx} [/Tex].
Solution:
[Tex]I = \int^2_1\int^4_3 (2x + 3y)~dy~dx [/Tex]
[Tex]\Rightarrow I = \int^2_1[\frac{2x^2}{2}+3xy]^{x=4}_{x=3}dy [/Tex]
Putting the upper and lower limits for x, we get
[Tex]\Rightarrow I = \int^2_1[\frac{x^2y}{2}]^{x=4}_{x=3}dy = \int^2_1[4^2+3(4)y-(3^2+3(3)y)]dy [/Tex]
[Tex]\Rightarrow I = \int^2_1[\frac{x^2y}{2}]^{x=4}_{x=3}dy = \int^2_1(16+12y -9-9y)dy = \int^2_1(7+3y)dy [/Tex]
Now integrating w.r.t y, we get
[Tex]\Rightarrow I = \int^2_1[\frac{x^2y}{2}]^{x=4}_{x=3}dy = [7y+ \frac{3y^2}{2}]^{y=2}_{y=1} [/Tex]
[Tex]\Rightarrow I = \int^2_1[\frac{x^2y}{2}]^{x=4}_{x=3}dy = 7(2)+\frac{3(2^2)}{2}-(7(1)+\frac{3(1^2)}{2}) [/Tex]
⇒ I = 14 + 3 – 7 – 3/2 = 17/2
Example 3: Calculate [Tex]\bold{\int^1_0\int^4_2 x^2y~dy~dx} [/Tex].
Solution:
[Tex]I = \int^1_0\int^4_2 x^2y~dy~dx [/Tex]
[Tex]\Rightarrow I = \int^0_1[\frac{x^3y}{3}]^{x=4}_{x=2}dy [/Tex]
Putting the upper and lower limits for x, we get
[Tex]\Rightarrow I = \int^2_1[\frac{x^2y}{2}]^{x=4}_{x=3}dy = \int^1_0 (\frac{64y}{3}-\frac{8y}{3}) dy [/Tex]
[Tex]\Rightarrow I = \int^2_1[\frac{x^2y}{2}]^{x=4}_{x=3}dy = \int^1_0(\frac{56y}{3})dy [/Tex]
[Tex]\Rightarrow I = \int^2_1[\frac{x^2y}{2}]^{x=4}_{x=3}dy \frac{56}{3}\int^1_0y~dy [/Tex]
Now integrating w.r.t y, we get
[Tex]\Rightarrow I = \int^2_1[\frac{x^2y}{2}]^{x=4}_{x=3}dy = \frac{56}{3}[\frac{y^2}{2}]^{y=1}_{y=0} [/Tex]
[Tex]\Rightarrow I = \int^2_1[\frac{x^2y}{2}]^{x=4}_{x=3}dy = \frac{56}{3}(\frac{1}{2}-0) [/Tex]
⇒ I = 56/6
Example 4: Calculate [Tex]\bold{\int^2_1\int^4_3 2xy~dy~dx} [/Tex].
Solution:
Using [Tex]\int \int_R kf(x,y)dy~dx = k \int\int_R f(x,y)dy~dx [/Tex]
[Tex]I = \int^2_1\int^4_3 2xy~dy~dx [/Tex]
[Tex]\Rightarrow I = 2\int^2_1\int^4_3 xy~dy~dx [/Tex]
[Tex]\Rightarrow I = \int^2_1[\frac{x^2y}{2}]^{x=4}_{x=3}dy = 2\int^2_1[\frac{x^2y}{2}]^{x=4}_{x=3}dy [/Tex]
Putting the upper and lower limits for x, we get
[Tex]\Rightarrow I= 2\int^2_1[\frac{4^2y}{2}-\frac{3^2y}{2}]dy [/Tex]
[Tex]\Rightarrow I= 2\int^2_1\frac{7y}{2}dy [/Tex]
Now integrating w.r.t y, we get
[Tex]\Rightarrow I= 2[\frac{7y^2}{4}]^{y=2}_{y=1} [/Tex]
[Tex]\Rightarrow I= 2[\frac{7(2^2)}{4}-\frac{7(1^2)}{4}] [/Tex]
⇒ I = 2(7 – 7/4) = 21/2
Example 5: Calculate [Tex]\bold{\int^5_0\int^6_0 x^3y^2~dy~dx} [/Tex].
Solution:
Let [Tex]I = \int^5_0\int^6_0 x^3y^2~dy~dx [/Tex]
[Tex] \Rightarrow I= \int^5_0[\frac{x^4y^2}{4}]^{x=6}_{x=0} [/Tex]
Putting the upper and lower limits for x, we get
[Tex]\Rightarrow I= \int^5_0[\frac{1296y^2}{4}-0]dy [/Tex]
[Tex]\Rightarrow I= \int^5_0\frac{324y^2}{2}dy [/Tex]
[Tex]\Rightarrow I= 162 \int^5_0 y^2dy [/Tex]
Now integrating w.r.t y, we get
[Tex]\Rightarrow I= 162[\frac{y^3}{3}]^{y=5}_{y=0} [/Tex]
[Tex]\Rightarrow I= 162[\frac{125}{3}-0] [/Tex]
⇒ I = 54(125) = 6750
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.
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