By direct integration determine the coordinates of the centr

By direct integration, determine the coordinates of the centroid of the trapezoidal area.

Solution

Solution is as follows

We need to set up intergration as dydx

The lower bound of intergration for dy is simply y=0.5x and the upper bound of intergration is y=0.2x+9. The bounds of intergration for dx are simply 0 to 9 and for dy 0 to 7

Now you did not state whether it has a variable density or not so I will write two different integrals.

first constant density m=p double integral dy dx

y=0.5x..0.2x+9 ,x=0....9 y=0...7

x=My/m y=Mx/m Mx=bounds same as above My=bounds same as above

second variable density m= double integral p(x,y) dydx bounds same as above the equations for My and Mx are the same as above except you add p(x,y) into the integral.