First, watch this Khan Academy Video where Sal discusses how to integrate using polar coordinates. In particular, he shows that the differential element of area in polar coordinates is dA=12r2dθ. Then prove, using integration with polar coordinates, that the area of a semi-circle is πr2 and that ˉy=4r3π.
Hints:
- The equation of a circle with radius r in polar coordinates is r(θ)=r.
- The process to find area and centroids is exactly the same as previous problems, only the form of the integrals is different.
- Finding the area of a circle using polar coordinates is trivially easy.
- To find Qx you will need to determine ˉyel as a function of r and θ.
