Question

Hey Guys

#50points

Find the area of region bounded by a semicircle whose diameter is 14 cm and the centre is at origin and the lines y = - x + 7 & y = x + 7 . ( semicircle lies above the x - axis )

Answer 1

I hope this answer help you

Answer 2

Let length of the side be x, Then the length of the other side is 2r2−x2−−−−−−√

, as shown in the image.

Rectangle inscribed in a semi circle http://i61.tinypic.com/2m7vu41.jpg

Then the area function is

A(x)=2xr2−x2−−−−−−√ A′(x)=2r2−x2−−−−−−√−4xr2−x2−−−−−−√=2r2−x2−−−−−−√(r2−2x−x2)

setting A′(x)=0

,

⟹x2+2x−r2=0

Solving, I obtained:

x=−1±1+r2−−−−−√

That however is not the correct answer, I cannot see where I've gone wrong? Can someone point out any errors and guide me the correct direction. I have a feeling that I have erred in the differentiation.

Also how do I show that area obtained is a maximum, because the double derivative test here is long and tedious.

Thanks!

One method of obtaining this equation is to integrate dxdy over the area of a circle. Well, you probably wouldn’t want to do that in cartesian coordinates – but you get the idea. I recently saw a graphical derivation of the area of a circle. Let’s say you start with a circle and break it into 4 wedges. The area of the 4 wedges should be the area of the circle