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Evaluate the following integrals in cylindrical coondinates. The figures, if given, illustrate the region of integration.$$\int_{0}^{3} \int_{0}^{\sqrt{9-x^{2}}} \int_{0}^{\sqrt{x^{2}+y^{2}}}\left(x^{2}+y^{2}\right)^{-1 / 2} d z d y d x$$

$9\pi/4$

Calculus 1 / AB

Calculus 2 / BC

Calculus 3

Chapter 16

Multiple Integration

Section 5

Triple Integrals in Cylindrical and Spherical Coordinates

Limits

Integrals

Integration

Integration Techniques

Multiple Integrals

Integration Review

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so and this question were asked to be very rate this triple, which is a triple integral of X squared plus y squared to the power of negative half. Where's he goes from? Zero to the square root of X squared plus y squared. Why goes from zero to the square root of nine minus X squared. Expose from zero. Okay, now what do we first do? So since we're going to cylindrical coordinates, we know that expert. That's why square it is important for square. So this right over here in this bar square. So we have our squared to the power of negative one hand. But also here we have expert plus y squared, which is also R squared and the square root of our swear it is just part. So our C goes from zero to our Now we know that our why goes from zero to the square root of nine minus exploring our X goes from zero. So I tried toe brought this. So this is my ex. This is a lie. So first we have that expose from 02 threes of I plug in zero for the square root of nine minus x squared. That means that zero I get square root of nine, which is simply three. And now, when excess three law is going to be the square root of nine minus three squared or zero. So basically, I have this border over certain. So we see that our why goes sorry are our boats prophecy. Oregon three Vilar Beta goes from zero to Firebirds. So next time you have the are we can see that our our goes from 0 to 3 by large FINA goes from 0 to 5/2. All right, now, just one extra thing. This volume element to go from rectangular two cylindrical port. It's it's are easy artist. So we have to add this extra are okay now are square to the power of negative 1/2 twos. Cancels are left with our to the negative one Now are to the negative one times ours one. So, basically, are you triple integral? Is this right over here? So now what's the integral of, uh, easy? Hold us to see where's the ranges from? Zero are. So if we do, our minus your we just get on. So now we have to do this double integral over here. So the double integral of our where argo, where are goes from 0 to 3 and data goes from 0 to 5. Now, the integral of are the oranges are squared, divided by two are limited integrations from Syria. So if we plug those and we get nine square, give a three swear divided by to buy This year of squared divided by two which is just nine so nine have the state nine has is a constant. So now we think the integral with respect data's just gonna be nine house time state up where if they don't goes from zero to pile with too. So they plug that into get 9/2 times higher for to my zero, which is simply nine pie divided by four. So the value of our triple integral is nine pie divided by four

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