square $ [3,5] \times [1,4] $ theorem Green

Question

I need to use a line integral for the square with vertices $(3,4), (5,4), (5,1), (3,1)$, using the green theorem i have $\displaystyle\int_3^5-\frac{15}{2}(4x^2+3)dx =-30\int_3^5 x^2dx-\frac{45}{2}\int_3^5 dx=-980-45=-1025$ which according to the book is correct, but by means of parameterization of the square I don't get the result. How would the integrals be 4 parameterizing each line $ x = 3 $, $ x = 5 $, $ y = 1 $, $ y = 4 $ in an anti-clockwise direction?

Answer 1

For example, consider the edge between $(a,b)$ and $(c,d)$. You can parameterize this by $(1-t)(a,b)+t(c,d)$ where $0 \leq t \leq 1$. For example, consider the edge between $(3,1)$ and $(3,4)$, i.e, line $x=3$ in your setting. Since the direction is counter-clockwise we will start from $(3,4)$. Then, $(1-t)(3,4)+t(3,1)=(3,1-3t)$. Now observe that when $t=0$ we have $(3,4)$ and when $t=1$ we have $(3,1)$. What does $(3,1-3t)$ means is the following: You will put $x=3$, $y=1-3t$ and take the integral with respect to $t$ between 0 and 1.