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Consider the polar function: "(o)sin( 0). sin(0)0 < 0 < # (0), { <0 <(a) Graph ,(0) (b) Compute the exact value of the area enclosed by the curve...

Question

Consider the polar function: "(o)sin( 0). sin(0)0 < 0 < # (0), { <0 <(a) Graph ,(0) (b) Compute the exact value of the area enclosed by the curve

Consider the polar function: "(o) sin( 0). sin(0) 0 < 0 < # (0), { <0 < (a) Graph ,(0) (b) Compute the exact value of the area enclosed by the curve



Answers

Use a graph to give a rough estimate of the area of the region that lies beneath the given curve. Then find the exact area.

$ y = \sin x $,


$ 0 \le x \le \pi $

In this problem we have to evaluate the area of the curve. R equals A times the co sign of data from negative one half by to 1/2 by Now the area of a polar curve is given by the Formula 1/2 The integral from A to B. Of R. Squared D. Theater. And we know that the function R squared is even and the limits are equal except the ones negative. So we can actually have are integral. Going from negative one half by To one half by of our square data as being equal to 1/2 dimes to uh huh of the integral going from zero to buy by two and the two and one half cancel out. And we get to the integral as 0 to 1 half of R squared D. Theta. From here we can just plug in our as eight times the coastline of data and we get a squared times the cosine squared Hey to the theater we can pull out the A square from the integral because it's a constant and we get the integral from zero to buy of cosine squared of data. Data. Data from here we can use the identity that go sign square with data equals one plus the co sign of two. Theta divided by two. We get the integral going from zero to buy by two. This was a bye bye to of one plus go sign of two. Theta divided by two. From here we can do the integration. We can pull the two out of the integral. The integral of one is just data with respect to data and the integral of cosine of duty to will be the sign of two. Theta divided by two going from zero Dubai by two. From here. We can just substitute in our limits and we get bye bye to plus sign of two times by by two Divided by 2 0 plus sign of two times 0, divided by two. And this entire thing will evaluate, do a squared times pi, divided by four, which is a required area.

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.

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.

Okay. Using the gruff first, we show that this is our shaded region. Over here, we know this goes from 002 pi comma zero. Which means that if the area which square is 1/4 time's 1/4 which is 1/16 that we can calculate the area to be approximately the number of squares. So 60 two ish divide by 14 times 1/4 which is 1/16 which is like 3.9. Okay, let's calculate the exact area now. Zero pie. I noticed. These are our bounds on the graph top minus bottom. You know, the integral of sine X Executive Co Cenex. The two goes at the top. We pull out the two that's imprint the season. Make it 1/2 on the outside, the integral of Sinus coastline. This plugging in fundamental there of calculus. This gives us four. This is really close to 3.9


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