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I want to find the volume under the slanted plane

$$f(x,y)=\frac{101}{1448}x + 56 $$

bounded by a circle with radius 7.7. I've used all the rules in polar coordinates to try and find this volume, but when I do it I get a volume defined by a cylinder with formula

$$ V=\pi r^2 h $$ where $h=56$ and $r=7.7$.

-- an IDENTICAL answer when they SHOULDN'T be the same since that plane is slanted and that cylinder assumes a horizontally flat top, they are different. They should be similar but NOT THE SAME. What am I doing wrong??

Is it because I can only work with symmetrical functions with polar coordinates?

EDIT: can someone answer the question with a formula in purely double integrals that can find the volume under that slanted plane?

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    Why do you want to use polar coordinates? – 2017-02-22

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If you are bounded in the plane by the disk $x^2+y^2 \leq (7.7)^2$ then this is your domain. You can set up the triple integral in cylidrical coordinates as follows:

$$\int_{0}^{2\pi} \int_{0}^{7.7} r \int_{0}^{\frac{101}{1448} (7.7 \cos \theta) + 56} 1 \ dzdrd\theta$$

enter image description here

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    Does it have to be a triple integral? I cannot do a simple volume like this using a double integral? – 2017-02-22
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    The problem is, the top is slanted. What you can also do is find the volume of the cylinder with radius $7.7$ and height $56$ then subtract off from that the excess volume given by: $$\int_{0}^{2\pi}\int_{0}^{7.7} r \int_{\frac{101x}{1448} + 56}^{56} 1 \ dzdrd\theta$$ which seems like a lot of extra work. – 2017-02-22
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    Is the double integral $$2 \int ^\pi _0 \int ^{7.7} _0 (\frac{101}{1448} (r cos \theta ) +56)\, r \, dr \, d \theta $$ not sufficient? – 2017-02-22
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    thanks very much - how would you parameterise the region for a double integral? – 2017-02-22
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    That's exactly the same integral. The only step that you've simplified is my first integral which is your first integrand. And the object is not symmetric about the x-axis, so you can just double, so change the ending integral limits back to $0,2\pi$. – 2017-02-22
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    I evaluated that double integral and it gave me teh same answer as the formula for a cylinder would. I even checked it on double integral calculators. Why is that? I got 10404.8, which is $\pi r^2 h$, cylinder volume, which is wrong. the double integral should be slightly less than a full cylinder because of the slant. how is this possible? (height is actually 55.86 but i rounded up in the question. if you use this height for the integral and for the cylinder formula theyre equal somehow?) – 2017-02-22
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    This is not the volume of a cylinder! The top is slanted. Of course you would get the same answer as the volume of a cylinder if you set the integral up as if that's what you wanted to find. Your integral calculates the volume of a cylinder of radius $7,7$ and heigh $56$ because you assume that a symmetry is at the $y$-axis. This is done when you wrote $$2\int_{0}^{\pi} \cdots$$ – 2017-02-22
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    i see, how would I not assume symmetry? how to do it with double integrals then? do i set the boundaries of $\theta$ as $[0,2\pi]$ ? – 2017-02-22
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    Let us [continue this discussion in chat](http://chat.stackexchange.com/rooms/54069/discussion-between-m98b-pro-and-faraad-armwood). – 2017-02-22
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    You don't need a triple integral, however it would have started out as one, if you notice the innermost z integral easily evaluates to the function defining the slanted top of the cylinder, which you could have started with in the beginning as the integrand for a double integral – 2017-02-22
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    so can anyone answer the question with a formula in double integrals that can find the volume under that slanted plane? – 2017-02-22
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    No @Faraad, the problem is symmetric about the plane $y=0$ and, I presume because chance, it's reflected in the factor 2 and the **limits of the integral** – 2017-02-22
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    @RafaBudría, yes that is true of course, but the OP goes from $[0, \pi]$ when you should go from $[-\pi/2, \pi/2]$ and double! – 2017-02-23
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    @Faraad No, no. $y$ and $-y$ get the same value for $f$. He get the same volume as for a cylinder is because the slant is of no effect at all!! – 2017-02-23