Exercise 5.3: Consider the integral Elr) = /" e-"147CHAPTER 5INTEGRALS AND DERIVATIVESWclte program (o calculate E(x) for values of Irom to 3 in steps of ...

Given the definite integral $$ I=\int_{0}^{3} e^{-x} \cos x d x $$ (A) Graph the integrand $f(x)=e^{-x} \cos x$ over [0,3] (B) Use the right sum $R_{6}$ (see Section 5.5 ) to approximate $I$

Yeah. The black curve in the graph is a functioning FX equals be to empower minus X times co signing acts. And for part B we want to use the rice up to approximate this integral. That is. We just use the area of the six rectangular to approximate the sentry group and they know that the white of all the six rectangular czar point of fuck in their hearts. The function value are these sticks points, respectively, Enhance the some of the error of this. six rectangular Equals .5 times the sum of their function values and to evaluate their function values. Radio calculator and then the final answer is approximately .288.

Okay. In discussion we have to solve integration of areas to the power minus three to cost to duty to by using the table of integral. Okay so we will go through the develop integral and we will find part to formula number nine. That can be applicable in our case. Okay. And the formula is integration of a raise to the power X. Cause B X D X. And that is close to one divided by a squared plus B square. He raised to the power eggs into the bracket. A cost B X plus B. Sign bx plus C. Okay. And now we will apply this formula in our case and while comparing to this we will get the rest of the power X. In our question it is the rest of the government of ST rita. Okay so here value of Is -3. Okay. And Cosby X in our question there is question to be easy equals to one. Okay. And we will put X equals to twitter and the X equals two digit to. So It will be integration of E raised to the power -3 to to course Kita did you to Okay using this formula where it will be one divided by a squared plus B squared. So it will be one divided by minus three square plus one square. Here's to the Bowery X. So it will be erased to Devour -3. Okay here three days variable. No tax and now into the bracket Air Cosby X. So it will be A is minus three and cause be twitter, B is one so it will be to it only and plus B sign B. X. So B is +11 sign twitter. Okay and plus constant now we will simplifying it. It will be integration of us to give our minus three twitter cost you to do to to it is one divided by it will be minus three square that is nine and plus one is square. That is one. So nine plus one. It will be 10 in the denominator. He raised to the power minus +32 to Okay and into the bracket it will be minus three cost data plus scientist to. Okay and plus C. Okay now we can expand it. So it will be mhm Integration of the rest of the power -3 Twitter cost to to deter equals two. It will be this whole person will be multiplied by this and this. Okay so it will be minus three by 10. He raised to the power minus three to to cost you to Okay and it will be plus one by 10. He raised to the power minus three to scientific to plus C. Okay and this will be the answer of this question. Thank you.

Okay, we have an integral here and we're going to use its graph to estimate the actual area underneath the cur and therefore the answer to this integral problem. Okay, so how we're going to do that is we're going to use our average value. Now, our average value will be the line that we draw that has the same amount of area above that line as below. So it's kind of like taking that top part and filling it into the other parts. This creates a nice rectangle that's much easier to find the area. So I'm going to consider that height to be um 3.3. It's kind of hard. You know, you're you're going to find that your values are going to be a little bit different. Someone might have used anywhere from 3.2 to 3.4 And then our base is too, so we have a height time space. So I'm going to say that our estimate is 6.6 and again you can be within a range of that value.

So let's take control of this integral right here. This is a definite in a croquet. Still gonna use the integration by parts. Everything that we've done so far under an addition by Parsons gonna work. Your thing now is that there's not gonna be any policy at the at the end. It's just gonna be, uh I put a limit on this newer Nimet did before integration by forest thing. Okay, so I really ate. It's gonna help us. So the problem. So you're just gonna let you okay, eggs? Because eggs ology break and exponential is gonna be our TV. No one is gonna be our d u r d u is t x to use one source just e x. Okay. And he is gonna be the general off this famous guy right here. And that is gonna be Eataly. You know, you ve you know, Now you have This is the equivalent, uh, definite in a row for an admission by parts case. You ve it to be easy. You So this is the same this time. We just evaluating it, um, at upper an ornament. And this time, our upper, an ornament or zero and three and zero, respectively. Okay, So you ve r you is x r v iss uh, Thio. He's a negative Easy t x or two. And then our parliament as three and honor element is zero minus integral 0 to 3. V d u r v is too extreme. Do you eat it? Excellent, too. So I'm just gonna put out too. And then, uh, e thio excluded too. And aren't you? It's just the X. Okay, Now we have to do one more integration and this dude right here, uh, Sue, this one is gonna be thio 03 This the ex This is actually gonna be, uh, again too. Okay, then Thio e t. Excellent too generated. Three. This wish you have. So we're just gonna replace this one? Bite that on a different page. So we have two ex eatery expert you 03 my ass. Uh, you know this Is that so just gonna before? Can you see that? Two times two is four citizens guy before e extra pretty Also 03 So, this one, I probably many matters. Laura Limits. That is gonna be two out. Two out, two out. And you have a problem it off three e 232 minus. No, really. You know, when I put zero here, everything is gonna be see, Rose. So I can I can forget about that room and just bring this one right here like that. That minus same thing's gonna happen here if I know it's not gonna happen. So if I have this, this is gonna be e to three over to, you know, I'm put in effort. Limit in place of X and minus eats at the lower limit. So zero with you is what I have. So this is actually gonna be, uh do you can pick your candidate er's and you have. Ah. So you have fix, because this is two times three as six k 76 each of the three over two and you know, zero eater zero. This is gonna be zero. So any number two powers Series one. So this is gonna be negative for E to this three. Do you know this one says good positive for So you have this one thio do you with So you can just take your car clearer and put it in. Put it in. So this is gonna be, uh, six and then for e to the three over two. So when you put this one in your concrete already, you have approximately approximately, uh, you know, approximately 12 0.96 So that is that's what you get in. So, uh, the integral of this one from 0 to 3 iss approximately. I beg your pardon? Eso the interval? It's approximately 12 point. Nice six.