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Evaluate the given integral.$$\int_{0}^{1} \int_{0}^{2}\left(x^{2}+y^{2}+2\right) d y d x$$

$$22 / 3$$

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 6

Double Integrals

Partial Derivatives

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Lectures

12:15

In calculus, partial derivatives are derivatives of a function with respect to one or more of its arguments, where the other arguments are treated as constants. Partial derivatives contrast with total derivatives, which are derivatives of the total function with respect to all of its arguments.

04:14

04:42

Evaluate the given integra…

02:51

Evaluate.$$\int_{0}^{1…

02:35

Evaluate.$$\int_{0}^{2…

01:19

Evaluate $\int_{0}^{2} \ma…

06:19

01:37

Evaluate:

02:41

Evaluate

00:17

Evaluate the definite inte…

you know in this video I'll show how to integrate this function we're here so we have X two or two and spite report two plus two and then we have new ideas so I would approach this double instagram is firstly we need to integrate this in a part with respect to y as the dy so remember when we are integrating with respect why everything else which is not a Y in yet malfunction. We treat it as a constant. So not only treat the constant but also extra portrait constant when we're integrating with respect why? So firstly You need the integral from 0 to chew X squared plus y squared sense of life plus two and then that's a bit too why? So the interview of this we know that X squared is a constant. So we're going to have big squid. Why? Yeah class the integral of y squared you know that's gonna be on the two or three over three. And then lastly we have to why and these Yeah I play all boundaries which is from zero and two. Okay great. Now replacing our boundaries we know that please. So we remember don't be confused here and we are placing in position of Why? So the first one is going to become two X squared plans so we have 2 to the power three three. We have to So you were two years So we won't write zero because everything else is going to become the they were there. So proceeding with this, you're going to have to dates of departure and then we have 2 to the power three which is eight And then so its size 813. Then lastly we have plus four. So here you can evaluate this eight over three plus four during that calculation so 8/3 plus four begin to 20 or three some here to expect it 20/3. Great. So this is the integral of the inner point. Now we need to find the outer intake law from 0 to 1 with respect to X. We're integrating this function which we Found here so from 0 to 1 and you make integrating two X wed plus 20/3. And remember now this is with respect to X amount so the X proceeding with that we know that first part on the integral is going to be Tricks .3 Over three place 20 or three X. And then again we'll play our boundaries From 0 to 1 And then evaluating this so you're replacing 1% of X. So the first one is gonna be two, 1 to the poor three 43. Close 20 over three for more playing it with month. again I won't write zero because separating with placing zeros we just want to have a show there So we have two or 3. Class Twin. The over three. And this within is me. And since we have common denominators so we're gonna have 22 over three so this becomes the doubling technology. Oh fuck function. We had about thank you for your time.

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