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

Yeah, this black curve is a graph of the function F. X equals eat a power hour minus X. Time science. And for part B we want to use the left some to approximate this integral. And here we just use The area of this six rectangular to approximate this into. It's easy for us to see that the white of the six rectangular, all part five and their heights as a value of the functioning and zero point of five, 1 1.5, 2 and 2.5 respectively. Now in the hands, the some of the area of the six rectangular czar, the white, Which is .5 times the sum of their function value respective. And then use a calculator. We can find the value, Which is approximately .498.

In problem treat We have to evaluate the definite integral for two mm six D. X. Using the limit definition. So for this. Yeah. The function if have X. E. S equal to suits. Now we find lift of each rectangle which is delta X. Is equal to be my brother. Mhm. Yeah the upper limit of the definite intervene Stand minus is lower than it is for you are right. And risk gives The wrist of each directing you is six divided by mm. No choosing the right influence of the sub internment. We were see I is it going to? Yeah. Nice. Five times darkness. Yeah, it's the lord which is for less. I don't think the diet is six vitamins. So we would sit by do I buy. Yeah. No bye definition of the limited use not definite individual. We have particular area up to be yes D. S. It's equal to they meet and approaches infinity submission I from one up to N. Yeah. See I right delta. Yes. Uh huh. This is equal to mm hmm. And it works is being community submission. I've run up the end If uh see you see I man museum have CIA in the given countries we get F. R. C. I. is equal to six and then X. Is by mm So it will be a limit and it goes to school. Can you do from this year? And I'm from one of the wings At eight seats of I. Mm So unit and approaches to infinity Back to the six. Right mm. Outside destination. This year we get ice from one to the end of Yeah. And this is going to end so you get unit. And that brought us to infinitely 36 by N. Drains mm and your concern. And we grew up limit, then approaches to infinity 36 which is going to texas. So this is the require integral.

In problem. 27. Given this definite integral, we want to grab the integrity. Every X equals E. Is about minus X. So in X over From 0-3, which is this board of the integrity. This full body to graph F of X. We need to substitute by different values of X in the function and get its corresponding value. Let's start by x equals zero and get a full backs. When X equals zero, The function is it is a lot of zero which is one multiplied by 7.0, Give zero. Then fine answer is zero. You put half X equals half. Then we head into the bottle minus half, multiplied by signing off. And here sign is in radiance. Then when we calculate this value using calculator, we should make sure that our calculator is in the region moods. When you substitute by hove, it gives open tonight. Can I submit one in this small place? When you substitute by x equals one. It gives a .3. What? When we substitute by X equals 1.5. It gives all point to two. Ministers were x equals two. It gives oh 0.1 two. When you substitute by x equals 2.5. It gives all .05 And find anyone some super x equals three. It gives almost zero because we're very close to poi and signed by equals you. It's almost zero. To graph it. The x axis starts from 03. Then we have a year one, two, three And the halfs. And for the access Its maximum value is over in three. Then we can get step of all .1 oh 0.1 Oh boy in two oh 0.3. And the first point is at zero and 0. Second point is at half and open three year. The 3rd point is at one and 4.3. Um The 3rd point, the 4th point is 1.5 and open in two years. Then two and a .1 here 2.5 and 4.05 year. And finally We have 3 n0 here. We can see that we have here a big then goes something like that. And this is a draft of F of X. How about we? We won't use the left some L six to approximate. Oy you want to approximate oil. The forgiven definite integral using the left some that is that if some of six means we have six segments of the grave, the first one starts at X equals zero, which has a value of zero. Then the first rectangular he's here. You can put it with a different color. This is the first area. Second area starts at 4.5- one and has a value of 4.3. Then this is the second area. The third area is at one From 1 to 1.5. With the value of over 1 3 31 starts at 1.5 as the value of open to and so on. These are the six segments that approximate the function of X. When we get this area, the area of the blue rectangular, we got an approximation to the area under X. Which is equivalent to the definite integral. Then it equals The first area is zero plus the second area Is the height multiplied by the bet 0.3 blind by the weight. 0.5 plus the third area As the height of 4.3 and with 0.5 plus all point to was deployed by the with all has the same with plus a 0.1. Applied by the earth plus Or .45 multiplied. Boy, though it's these are the second segments that approximate the integral equals 4.475. And this is a final answer of our problem.

In the given questions were able to integrate it within the limits of 02 Bye bye to Okay The limits are able to play by two and the function is Synnex into FDS course EX or Dx. Ok. So to start with diet saturating feel first of all multiplex of 16 then we get minus sign Xv xvi Close to duty means cynics dx is substituted by minus stadium cape. Now for the new variability to find the new limits here because the old limits that have zero and a way to stand for X. So when it's zero or when X is by by two then we have to calculate the value of penalties for sex. So close on zero is 1. And that test case because of my way to which is a little okay by law substitute this substitution. Then I can say that the new limited that went to zero and at the place of sine X dx we have to put minus dating and here use a party with dignity. Okay this changes into- or went to zero. Have their stated So we have to integrate our derivative. And we know that integration. It's already time to derivative. Okay. You know that the basic thing says that it is empty very voting. So it's you. Integrator Derivative then you should get it into 20 derivative means the real fortune. That was okay. So here I should get the function that was. So finally I can say that the function that is a take and now we have to put your limbs. I put your upper limit zero when I put your lower limit to one. Okay Now this table is helpful to find the value of zero that is five And we love that what? That is seven. So here we have to do- of 5 -7. And we get finally to okay, this is the final answer here. That is too. Okay, thank you.