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Evaluate the given integral.$$\int_{0}^{4} \int_{x}^{2 x}\left(x^{2}+3 y^{2}+2\right) d y d x$$

$$528$$

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 6

Double Integrals

Partial Derivatives

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No my name is Butler. And then this video I'll show how to evaluate this integral. So you have a function which is a function of X. And Y. It's a two plus three white part two plus two. The I. P. X. And then we're also giving our boundaries. So firstly integrating at that point it girl. We first need to integrate this inner part. And then lastly we integrate the outer part. So member of the inner patriot with the line which means it is with respect why. And if you take a look in our function we also have X. And the to be sure not to fruit as a constant but then also X be treated as a constant because we are integrating with respect to why? So firstly we're going to integrate our enough but so we have from X. Two X. And then integrating a function. She's X squared. That's three white party plus two. And then it is with I respect to why? Okay. So we know that we have our concerns and our expertise. We're going to have X. Weird. Why? Plus Here we have three white poor too. We know that it becomes white or 3/3 and then the three years comes to each other. So we're gonna be left right to the part three plus two. Why? Then we have our boundaries X. Up to two X. Okay. Now are playing our boundaries plugging in? No you're plugging in on where we are. Why? So I'm gonna have X squared and then you have to X. And here we have two x 2.3 and then the last thing you have to True X. Okay. And then playing our lower boundary sweat, you're going to have X plus X to the 4th 3 and then lastly we have to X. Okay. Greed. Now simply find this, you know that 1st 1 becomes two x to the power three And then this one becomes eight X to the power three And this one becomes four x minus. So remember everything inside the square brackets, going to become a minus 6 to 3 -X. to the power of three minus two X. Okay great. Simplifying this further. So we have two X or three plus 88243 which becomes 10 next to the port three. and as a minor sticks for three minutes or three which means we are left with eight extra three and then if it is a little bit of plus four here plus four X -2X. Which means I'm gonna be left with class two weeks. So this becomes our in the integral. Now we need to find out integral. Take a look It's from 0 to 4 with respect to X. So yeah, from zero to fall. Do you have all functions eight X. 10 or three plus two X. Now we have with respect to X. Okay, continuing further with this we know that This is going to become eight x. The whole over four plus two x 2 or 2/2. And then we have boundaries yeah. From 0 to 4. Yeah. They're simplifying this. Free them here. We're gonna have to X to the poor form because X squared zero zero four. Okay. And then landing in here we're going to hear two spoiler. That's right. That for me please. Those are for These are bound to the call for plus four so they're quite two. Okay. Green Now we need to simplify that if we simplify that So we have four times so it's two times four to be powerful Plans for the two. Our end result becomes five too late. So this becomes a double integral. The full function. Thank you for you.

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