Question

Find the area of the shaded portion in the circle.

Answer 1

the total area is pi*6^2.From this we subtract the area of the strip.
We need to find the area of the strip.
To do this just consider the upper semi-circle and the sector which forms a 60 degree arc..So,
1/2 *pi*6^2 - 1/6*pi*6^2 + 2*1/2*3*3sqrt(3)
= 1/3pi*6^2+3sqrt(3)
Subtract this from pi*6^2 to get the grey portion.

Answer 2

im confused

Answer 3

Use the equation of a circle \[x^{2}+y^{2}=r^{2}\] with r=6 as the function to integrate so that you integrate with respect to y for the function \[y=\sqrt{36-x^{2}}\]
\[\mbox{Area of small white band}=2\times\int\limits_{0}^{3}\sqrt{36-x^{2}}dy\]

Substitute in dy from \[{dy\over dx}={1\over2}(36-x^{2})^{-{1\over2}} (-2x)\] by multiplying through by dx. Perform the integral to get -18 unless I'm mistaken. Ignore the negative sign because it's an area. Then subtract this from the total area of the circle of radius 6 via:
\[\mbox{Total area}=\pi r^{2}=36 \pi\]

P.S. I can't see the logic in the previous answer either I'm afraid so I couldn't help you there.

Answer 4

I meant, that from the semi circle subtract the of the sector.This sector has a central angle of 120 degrees as the radius is 6 and the width of the strip is 3.From this we can get the angle at the centre Now when you find the area of the sector, we need on the shaded portion hence everything below the chord that also forms the sector should be subtracted.These are actually 2 right angled triangles with 30-60-90 degrees.Hence the term
2*1/2*3*3sqrt(3) which is the area of these two triangles.

Answer 5

confused

Answer 6

these are the two triangles in the image, i'm talking about.

Answer 7

Yeah I just realised what you meant by the triangles before you posted that, but the rest of the area of that white band is bound by a curve, so you need to use an integral to take that curve into account.

Answer 8

OH! Ok no I see... lol

Answer 9

You treat those as sectors of the circle.

Answer 10

No that part can also be be got by the area of the sector of 120 degrees as well as the area of the semi circle.

Answer 11

yea exactly.