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Find the average value ofy =3x' + 2x over [-2,3]Find the area of the region in the 3" quadrant bounded by y = x andy =x....

Question

Find the average value ofy =3x' + 2x over [-2,3]Find the area of the region in the 3" quadrant bounded by y = x andy =x.

Find the average value ofy =3x' + 2x over [-2,3] Find the area of the region in the 3" quadrant bounded by y = x andy =x.



Answers

Find the average value of the function on the interval, using antiderivatives to compute the integral. $y=3 x^{2}+2 x, \quad[-1,2]$

For this problem, we're gonna be looking at double integral Zo ver general regions and this overall in our overarching concept of multiple integral. So in this case, we know that the average value of function is equal to one over a times the integral over the region. D of f of x Y d a. Andi A is the area of the region D. So in this case, we're defining the region d Aziz being zero less than or equal to y stirring equals X squared and zero less than or equal to x, which is less than equal to one. So now what we have is that our area is going to be equal to this double integral from 0 to 102 x squared de y dx. And when we saw that we're going to get one third because this would just be one right here, and then we would get X cubed over three. Evaluated at zero on one would just be one third. So with that being said, um, we now have our area. So when we do our f average, it's going to equal one over a So just three times the integral from 0 to 1 of negative x cosine Y um actually, we could go back further the double integral 0 to 10 x squared of X sign. Why de y dx. And when we evaluate this integral. But we're gonna end up getting through. Substitution is going to be three and then times one minus the sign of one over two and we could evaluate the sign of one. But overall, this is going to be our final answer by evaluating the double integral.

Okay. This question be able to find the area of the reason bounded by and the cars are why equals two three x divided by X -1 X -4. OK X -4. And why calls to zero X equals two and X equals 23 Okay, so these are the cars and we have to find out the area bounded by this car. So from this car we can say the area will be integration of why? D. X. Okay. And the limit will be from x equals 2-3. So it will be too okay. And three. So the area will be integration from 2 to 3 and it will be three X divided by x minus one X minus four. D. X. Okay. It means we have to find the definite integral off this integrated. So we will use the method partial fraction and three X divided by x minus one X minus four. That can be written as a divided by x minus one plus B. Divided by x minus four. We will take the L city in the right hand side and the numerator denominator will be same. We will compare the numerator three X equals to a x minus four plus b x minus one. And the denominator of both sides of the patient are same. So it will be cancel out. And now we have to find out the value of A. N. B. So uh X equals to one. Okay this will be zero and the left hand side will be three right and there will be a multiplied by minus three. From here it will be minus one and add X equals to four. This will become zero and we will get the value of B. So the left hand side will be three or four. That is 12. And writing style will be multiplied by 4 -1. That is three. So from here they will be four. Okay so three x divided by x minus one. X minus four. That can be done as divided by x minus one. So it will be minus one divided by x minus one. Will be divided by x minus four. So it will be four divided by x minus four. Again what we have to find out, we have to find out integration three X divided by x minus one X -4 DX from 2-3. Okay so it will be very simple integration from 2 to 3 of minus one divided by x minus one plus four divided by x minus four and D X. It means we have to solve this. So it will be integration Okay minus one divided by x minus one dx from 2 to 3 plus four. Integration one divided by x minus four D X. From 2 to 3. Okay and now very simple to solve. So it will be minus integration of one divided by x minus one. It will be Ln x minus one. Okay, It will be Ellen X -1. And the power the limit will be 2 to 3 plus four and it will be Ln x minus four. Okay and the limit will be from 2 to 3. So it will be minus and Ellen we will put a parliament minus Love elements. That will be three minus one. That is two minus Ln one plus four and now we will put same. So Ellen it will be three -4. That is -1 and minus Ellen it will be minus two. Okay and now it is minus Ln two and L n one is already zero. So it will be Ellen to minus zero and plus four. Okay, this is Model A. So it will be L. N. One and minus it will be a lesson to. Okay, so this island one is also zero and it will be minus Ellen too. Last four minus Ln two and the last it will be minus Ellen to minus four. Ellen too. And this will be minus five Ln to an area cannot be negative. So we will put Modelers area will be this. So the area will be five Ln two. Okay. And this will be the final answer of discussion. Five Ln two will be the final answer. Thank you.

We want to find the area bounded by Y equals four and y equals x two, the 2/3 in the first quadrant. This area right in here, so are left bound is going to be zero. And a right bound is gonna be where these two curves intersect. It will be at four equals x two, the 2/3. So it's so for X. Well, we can raise both sides to the third power four Cube gives a 64. The threes cancel and we're left with X squared. So X equals eight. And that will be our rate bound. So are integral. Goes from 0 to 8 and we're doing for minus X to the 2/3. And if we integrate, we get four x minus 3/5 x to the 5/3 and we'll pull you in eight and get four times eight minus 3/5 eight to the 5/3 and then minus plugging in zero, we just get zero. So this leaves us with 32 minus 3/5. Eight to the 5/3 is the same as saying eight to the 1/3 raised to the fifth Power and the cube root of eight is too, and to to the fifth is 32. So what we really have is 3/5 times 32. So are areas 30 to minus 3/5 times 32 which is going to be 2/5. Time is 32 or 64 over five, and that is our bounded area.

In the Russian. We have to find the average value off the function. F x home a white physicals minus excess one on the granular region with essays off 01 00 02 on board to Mama who now moving towards the solution. Yeah, F average is opposed. Toe double integral over the region B effects on my wife d do. I didn't buy double Integral was the region the from the exercise 79 with him to the conclusion that community that all over the region d efforts on why a is a close school to buy three so and the integration off double integral with the region thing The is the area off the shaded traveler Which is it going toe one by two into food and good food. This is for peace land and this is for the fight that is a closed toe. So the average and given us to buy three divided by a group that is close to 13 I doesn't finalize for shit. Thank you.


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