Solve a double integral given area, xbar and ybar?

[akjarvis0]

Homework Statement

If you know the area of a region with constant density, and you know xbar and ybar, then its possible to compute $$\int\int$$ ax+by dA for any constant a and b. [Hint: write down the formulas for the center of mass of a region.

If A=5 and (xbar,ybar)=(2,3), Compute$$\int\int$$ 7x-4y dA.

Homework Equations

xbar = double integral of x * density dA all over the double integral dA.

The Attempt at a Solution

I've tried using the area and the known values of xbar and ybar. but I really don't know where to get started. How can knowing xbar and ybar give us our bounds on the double integral?

 

[HallsofIvy]

So
$$\overline{x}= \frac{\int\int xdA}{\int\int dA}$$
$$\overline{y}= \frac{\int\int ydA}{\int\int dA}$$

From those,
$$\int\int x dA= \overline{x}\int\int dA$$
$$\int\int y dA= \overline{x}\int\int dA$$

And, of course,
$$\int \int 7x- 4y dA= 7\int\int x dA- 4\int\int y dA$$