Question

Show that the line integral $ \int\limits_{-1,2}^{3,1}{\left( {{y}^{2}}+2xy \right)dx+\left( {{x}^{2}}+2xy \right)dy} $ is independent of path and evaluate it.

Answer 1

Hint: We know that the integration of the equation for double integral will be variable based. Keeping others constant we integrate using the formula of $ \int{{{x}^{n}}dx}=\dfrac{{{x}^{n+1}}}{n+1}+c $ . Then we add them to find the final solution.

Complete step by step solution:
When we are integrating with respect to y all the other variables become constant.
Similar thing happens for when we are integrating with respect to x.
Therefore, we break the integration in two parts.
We complete the integration and then use the limits for the respective variables.
We get $ \int\limits_{-1,2}^{3,1}{\left( {{y}^{2}}+2xy \right)dx+\left( {{x}^{2}}+2xy \right)dy}=\int\limits_{-1,2}^{3,1}{\left( {{y}^{2}}+2xy \right)dx}+\int\limits_{-1,2}^{3,1}{\left( {{x}^{2}}+2xy \right)dy} $
For $ {{y}^{2}}+2xy $ , $ {{y}^{2}} $ and $ 2y $ will be treated as constant.
We also use the formula of $ \int{{{x}^{n}}dx}=\dfrac{{{x}^{n+1}}}{n+1}+c $ .
So, $ \int\limits_{-1,2}^{3,1}{\left( {{y}^{2}}+2xy \right)dx}=\left[ {{y}^{2}}x+y{{x}^{2}} \right]_{-1,2}^{3,1} $ .
Now we put the values for the variables and get
 $ \int\limits_{-1,2}^{3,1}{\left( {{y}^{2}}+2xy \right)dx}=\left[ 3+9+4-2 \right]=14 $
For $ {{x}^{2}}+2xy $ , $ {{x}^{2}} $ and $ 2x $ will be treated as constant.
We also use the formula of $ \int{{{x}^{n}}dx}=\dfrac{{{x}^{n+1}}}{n+1}+c $ .
So, $ \int\limits_{-1,2}^{3,1}{\left( {{x}^{2}}+2xy \right)dy}=\left[ {{x}^{2}}y+x{{y}^{2}} \right]_{-1,2}^{3,1} $ .
Now we put the values for the variables and get
 $ \int\limits_{-1,2}^{3,1}{\left( {{x}^{2}}+2xy \right)dy}=\left[ 9+3-2+4 \right]=14 $
Therefore, $ \int\limits_{-1,2}^{3,1}{\left( {{y}^{2}}+2xy \right)dx+\left( {{x}^{2}}+2xy \right)dy}=14+14=28 $ .
The line integral $ \int\limits_{-1,2}^{3,1}{\left( {{y}^{2}}+2xy \right)dx+\left( {{x}^{2}}+2xy \right)dy} $ is independent of path.

Note: The path integral of the equation is used for vector calculation. In qualitative terms, a line integral in vector calculus can be thought of as a measure of the total effect of a given tensor field along a given curve.