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Find the volume of the solid enclosed by the cone $z=\sqrt{x^{2}+y^{2}}$ between the planes $z=1$ and $z=2$.

$\frac{7 \pi}{3}$

Calculus 3

Chapter 15

Multiple Integrals

Section 7

Triple Integrals in Cylindrical and Spherical Coordinates

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Lectures

04:18

In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data. Integration is one of the two main operations of calculus, with its inverse operation, differentiation, being the other. The area above the x-axis adds to the total.

26:18

In mathematics, a double integral is an integral where the integrand is a function of two variables, and the integral is taken over some region in the Euclidean plane.

04:51

Cone and planes Find the v…

02:26

Find the volume of the sol…

So you're, following cone in 3 dimensional space of her the x, the y on the c axis, and we have this cone very nice cone described by the question. That c is equal to the square root of x, squared plus y squared. So we use cylindrical coordinates that t is equal to r and well. We want to complete what is the volume of this cone between making a cut through the plane c equals 1 and the plane c equals to 2 c equals to 2 point. So we would have a origion, like the waking between those 2 planes, so this plane, this lower plane is described by c equals 1 is only c is equal to 2 point, so would have this condition that we c is equal to r, but also c is Vain between 1 and 2, so we could make a or r to depend on c, because well here we're going to look at a transverse l like a profile. You would have these where your c is here, looking there 1 and 2 and then r go from 0 up to c. So r goes from 0 up to c this would be the r and then a c would be between 1 and 2. So this is for the volume and then our angle goes all the way around, because i want to obtain this figure this you take this around there the whole alter from 0 to 2 pi. So we want to integrate these while we have the interval of integral of a is equal to r square halves, so that this would be equal to during that c, squared halves, minus 0 point, and we write that, on the integral of the c, between 1 and 2 there, between 0 and 2 by well the traloc squared the zeros equal to c cuba, evaluate between 2 and 1, we're going to get 2 cubed minus 1 cube third, and this is equal to suechteight, 2 times 3 times 28 minus 1 point. So that is equal to 7 third, so this will get us our factor of 1 half and then these would be 7. Third, so 1 hue half times 7 third ether, and we can pull that out so that we have 7 third divided by 2 and then they grow from 0 to 2 pi ether and the draw for there is just which is going to be theta. Theta 2, pi and 0 all times that, and so this would be equal to 2 pi. So 7 times 2 pi divided by 3 times 2. This will be equal to 7 pi. Third, so it is that now 7 pi third seeby thirds. That is there. That is the volume of the volume of this region.

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