Integral over a parallelogram

Use the transformations
x=3s+t : \text{ and } : y=s+2t
to evaluate
$$\iint_P \left(x+2y\right)\,dA$$
over the parallelogram $P$ with vertices $(0,0)$, $(3,1)$, $(5,5)$, and $(2,4)$.


  • edited April 30

    The Jacobian

    We'll need to compute the Jacobian, that is, the determinant of


    Here's how:

    %%dx/(ds)=3, dx/dt=1, dy/ds=1, dy/dt=2%%


    Finding the bounds of integration

    Next, we'll find the bounds of integration:

    %%y=2x-5, y=2x, y=1/3x+10/3, y=1/3x%%

    %%s+2t=2(3s+t)-5 => s=1%%
    %%s+2t=2(3s+t) => s=0%%

    %%s+2t=1/3(3s+t)+10/3 => t=2%%
    %%s+2t=1/3(3s+t) => t=0%%

    Evaluating the integral

    Finally, we evaluate the integral:




Sign In or Register to comment.