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Question 10 (Fill-In-The-Blank Worth points)(09.05)Find the area enclosed by the graph of r = sin 40.Type your answer in the space below and give decimab places_ If...

Question

Question 10 (Fill-In-The-Blank Worth points)(09.05)Find the area enclosed by the graph of r = sin 40.Type your answer in the space below and give decimab places_ If your answer is less than place leading 0" before the decimal point (ex: 0.4821Answer tor Blank 1:

Question 10 (Fill-In-The-Blank Worth points) (09.05) Find the area enclosed by the graph of r = sin 40.Type your answer in the space below and give decimab places_ If your answer is less than place leading 0" before the decimal point (ex: 0.4821 Answer tor Blank 1:



Answers

The graph of the equation $r=\left|\sin 4 \theta \sin \frac{1}{2} \theta\right|, 0 \leq \theta \leq 2 \pi$, encloses a region that is a mathematical model for the shape of a horse chestnut leaf. See FIGURE 7.4.4. We shall see in Chapter 10 that the area $A$ bounded by this graph is given by $A=\frac{1}{2} \int_{0}^{2 \pi} r^{2} d \theta .$ Find this area. [Hint: Use one of the identities given in the instructions for Problems $47-52 .$

So if the following um set of problems, we're going to be looking to find the area um within this polar curve, the region bounded by the polar curve and what this is going to be very common formula that we're gonna be using in this section. That is equal to the integral from A to B. Of 1/2 r squared. I'm going to move the one half outside and then we have one half art squared the fatah. So then what that's going to end up showing us is that we know that our is going to be equal to the function with respect to data. So we get our, in terms of data in this case stated squared. So r squared would actually be fae to to the fourth so we can plug this in and we get um it should have certain power and depending on the equation that were given, will plug that in and then we'll also change the bounds of integration to give us our final answer. But these are the two equations that will allow us to solve any polar equation occurs to find the area bounded by them.

So if the following um set of problems, we're going to be looking to find the area um within this polar curve, the region bounded by the polar curve and what this is going to be very common formula that we're gonna be using in this section. That is equal to the integral from A to B. Of 1/2 r squared. I'm going to move the one half outside and then we have one half art squared the fatah. So then what that's going to end up showing us is that we know that our is going to be equal to the function with respect to data. So we get our, in terms of data in this case stated squared. So r squared would actually be fae to to the fourth so we can plug this in and we get um it should have certain power and depending on the equation that were given, will plug that in and then we'll also change the bounds of integration to give us our final answer. But these are the two equations that will allow us to solve any polar equation occurs to find the area bounded by them.

In discussion. We need to calculate the area of the region bounded by the given cubs which are vehicles to cynics and vehicles to one. And uh it is also given that X must be greater than equal to zero but less than equal to five x 2. So now, first of all we will plot the graphs of these curves so we plotted the graph of these two equations, vehicles to sin taxes here, this purple coffee and vehicles to one is this green straight line which is parallel to X axis. And the area between these cups is the shaded in blue. Now we can calculate this area by integrating five X minus geophysics. Were F of X? Is there a pack of which is why it goes to one and joe fixes. The local which will be very close to Synnex and the uh this region exchange from Y X equals to zero to accept close to piper to here. So we can calculate this area as the integration for the limit 0 to Pi by two. I know uh ffx minus jfx will be uh one minus this one is the pack up here when it comes to one is the podcast. So one minus a geo fixes. The cynics which is the lower cost. So this will be one minus Sine X. Nordics. Now the integral for this given expression can be calculated as the integral of one will be X. And the integral of cynics will be minus cosine X. So this will be Yeah Plus minus minus will live plus cosine X. For the limit 0 to Pi way too. Now by substituting their parliament, we will get area as ah private. Two plus. Consigned private to consign private will be zero. So here this will be only private tour. And for the lower limit it will be zero plus zero which will be ah Because I'm zero is 1. So this will be minus. Right? So we got the area As by by 2 -1 square units. I hope all of you got discussion. Thank you.

But if driving into salt from the number three Okay, here given thing is ask Clerical Tau nine side to fight artist much greater than or equal to 00 less than a record to fight less than a record toe private, half in a little Toby Ask where they fight Half in level 02 by the two man signed to fight different which is sick, or do nine by 20 to 5 to sign toe fight Do you fight so we can write life nine by to integrate it Be se knew Do you like substitute? We go toe to fight and the u equal toe the fight and the file equal to do you buy Took nine by four Integral aid Toby Se knew Do you? Yeah. Nine bite toe zero Goodbye Saying you d fight so lower bond you can like like Orban you equal to to window zero which is equal to zero apart from you equal to do in the pie by two which is equal toe so nine by five minus cause you zero took a bite. So land by four end of minus caused by minus minus called zero, which is land by four minus minus one minus minus one with just nine by four in tow. Physical toe, 18 by foot. This is a photo by thank you.


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