Question

Let R denote the circular region bounded by x^2+y^2 = 36. The lines x=4 and y=3 partition R into four regions R1, R2 ,R3 , and R4. Let [Ri] denote the area of region Ri. If [R1]>[R2]>[R3]>[R4] , then compute [R1]-[R2]-[R3]+[R4].

Could someone help me, I don't really get this...

Answer 1

Not sure how to do this without calculus; can't find any theorems about areas cut by intersecting chords.

So, consider the case where the lines x=h and y=k form the chords.

However, with calculus, you can "easily" show that R1+R4 = 1/2 pi r^2 + 2kh

That means that the rest of the circle, R2+R3 = 1/2 pi r^2 - 2kh

So, R1+R4 - (R2+R3) = 4kh

That is, the difference is just that of a rectangle 2h by 2k.

Answer 2

Thanks a lot Steve!!