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5 points) Compute TIL 22 dx dydz, where E = {(1,y,2) 12 +y2 < 1,0 < 2 < 1}....

Question

5 points) Compute TIL 22 dx dydz, where E = {(1,y,2) 12 +y2 < 1,0 < 2 < 1}.

5 points) Compute TIL 22 dx dydz, where E = {(1,y,2) 12 +y2 < 1,0 < 2 < 1}.



Answers

Evaluate the following integrals. $$\int_{0}^{5 / 2} \frac{d x}{\sqrt{25-x^{2}}}$$

Take a look at what we have here. You see that from X less than zero we got X squared So we're going to use that from negative to zero X greater tax. Never had the other one. So from 0 to 3 we're going to use negative x. So taking the answer during two of those We get next to the third power over three negative 20 S X squared over two. That's from 0 to 3, plugging those in. We get so we get cereal minus. So it needed to you Cute state of eight but minus the B plus eight thirds. Then the P -9 over to less cyril. Simplifying that out here. We've got 16/6 minus 18 27 actually over six. And so that simplifies out to 11 11/6 years. That's our solution.

Alright for this problem, the integral we're trying to approximate is the integral from 1 to 5 of x minus one. Both squared. Um And we want to approximated using midpoint rule with n equals two, then N equals four and then find the exact value by integration. So starting off we want to figure out our delta X. So that's going to be five minus one over two, first of all. So that will be 4/2 or just to which will give us uh Mid points Yeah, here the mid points will be one and three. So we just need to evaluate our our function at X equals one and at X equals three and multiply each by delta X. So our midpoint approximation for N equals two. We can obviously see that when we plug in X equals one, we'll just get a zero. Oh, excuse me. Uh That should actually be two and four. There we go. So we should have everything out front being multiplied by our delta X. So it's being multiplied by two, then we'll have a one second here. All right, uh Two minus one squared, so that's just going to be one plus four minus one squared. That's going to be three squared. So that's going to give us nine so of two times one plus nine. Um So that will give us 24 hour and equals two approximation and for the m equals four or n equals for rather our delta X is going to be just one, which means that our list of mid points here, it will be 3/2, 5/2, 7/2 and nine over to plugging in those points in evaluating our function, multiplying everything by our delta X, which is just going to be one, we should get that the terms are going to be 1/4 plus 9/4 plus 20 plus, 25/4 plus 49/4. Which when all summed up, we should get 21. Lastly integrating for integrating from one up to five x minus one. Both squared dx. So we can make the substitution that you equal to x minus one. So then that means that R D U is going to be unchanged except for our boundaries of our integration will be changed a little bit. If U is equal to x minus one then we'll go we'll be going from zero up to four of you squared the X or D you rather now. So that should give us U cubed over three, evaluated from zero up to four. So we should have four cubed over three which for cube is going to be 64. So we get 64/3 which is approximately 21.33333

Yeah. Alright for this problem we are asked to approximate the integral from 0 to 1 of E. To the power of negative X. Using the midpoint rule then to find the exact value. By integration. Uh We also want to express our answer to five decimal places. So we're integrating each of our negative X from 0 to 1 Where n equals five. So 1 2nd here we should have first of all that our delta X. What's delta X Is going to be just 1/5. So then our mid points Will be won over 10, 3/10, 1/2, 7/10 And 9/10. Having that we want to evaluate our function at each one of those points and multiply by delta X. So our N equals five midpoint approximation. It's going to be 1/5 times now we do our function evaluated at each one of those points. So we'd have each power negative 1/10 plus E. to the power of 3/10 plus course, sorry, negative 3/10 plus eat Power of negative one half plus E. To the power of negative 7/10 plus E. To the power of negative 9/10. And when we add all of that up We should get a final result of about 0.63, or actually seven if around in 25 decimal places. So next we want to actually find out what it is by doing direct integration. So the integral of E to the power of negative X is going to be negative E. To the power of negative X evaluated from zero up to one. So that is going to be let's see here. It will be negative E. To the power of negative one. Or negative one over E minus negative E. To the power of negative zero. That's going to be minus negative one. Let's see, daisy or plus one. So the exact value by integration would be 1 -1 over E, Which comes out to 1 2nd here. Uh 25 decimal places. That is 0.6321 two.

We're taking a look at the Inter Groll going from five halves to three and let me write down the Inter grand. Yeah, So we're gonna undo power rule to figure out the anti derivative 1/15. Mhm that we need to evaluate it. Three and 5.5. Yeah. So we have went over 15 and then we're evaluating at three. That gives us one to the 15. Minus zero to the 15. Gives us 1/15.


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