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Find the area of the region that is bounded by the given curve and lies in the specified sector=cos(0), 0 < 0 < 6...

Question

Find the area of the region that is bounded by the given curve and lies in the specified sector=cos(0), 0 < 0 < 6

Find the area of the region that is bounded by the given curve and lies in the specified sector = cos(0), 0 < 0 < 6



Answers

Find the area of the region that is bounded by the given curve and lies in the specified sector.

$ r = \cos \theta $, $ \quad 0 \leqslant \theta \leqslant \pi/6 $

He is clear. So when you read here So we have the area this equal to 1/2 the end to girl from A to B R squared Athena When we plugged things in, we get 1/2 been to grow from zero to Fada Sign plus co sign square deep data. This goes into 1/2 zero Fada signed Square Plus Co signed Square Plus to sign co sign he fada which is equal to 1/2. Then two girl zero data one plus sign to say tha d fada and this is equal to 1/2 data minus one have co signed to data from zero to pie which is equal to Pi house.

It's clear. So when you read here so we get a is equal to 1/2. Then to grow from a to B or a square D data, we substituted values to get 1/2 pie house to pie won over Data Square D data. This gives us 1/2 I have to pay to pie won over Data Square D Sita, which is equal to 1/2 negative one over theta from High House to two pi, which is equal to three over four by

He has clear. So when you read here, so we have our is equal to e to the negative beta over four. And we have data is bounded by pi over too, and pie included. So we have The area is equal to 1/2 then to grow a B or a square data. So if we plug it in, we got 1/2 pi over too. Hi. E to the negative data over four square data. This is equal to 1/2. They pi over too. Pie e to the negative data over to Encina. This is equal to 1/2 negative to e to the negative fate over to we have pie house pie. This is equal to e to the negative pi over four minus e to the negative pi over too.

For this given exercise we want to find the area bounded by the curve. So we have sine squared X. Mhm. Okay. Yeah and then we also have cubed And we're focusing on the interval from 0 to Pi. So looking here where X equals pi, this is the area between the curve that we're focused on. So it would be best if we had this value right here, the sign X squared minus the sine cubed dx. So we'll have the integral From 0 to Pi of sin X squared minus Synnex cube Jax. And we'll put parentheses around this whole thing. So once we evaluate this we get about 2.237 which is the same thing as one half pi minus four thirds. So that's our final answer.


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