How big is this polyhedron ?

Calculus Level pending

Let A A be the region bounded by the planes x = − 2 , x = − 1 , y = 2 z , y = 3 x , z = − x , z = 0 x= -2, x = -1, y = 2 z , y = 3 x , z = -x , z= 0 . Find the volume of A A . Given that the volume can be expressed as p q \dfrac{p}{q} for positive coprime integers p , q p, q , enter p + q p + q


The answer is 31.

Hosam Hajjir by Hosam Hajjir , 56, Canada

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1 solution

Hosam Hajjir
Dec 24, 2020

The volume is given by the triple integral

V = ∫ x = − 2 − 1 ∫ z = 0 − x ∫ y = 3 x 2 z d y d z d x V = \displaystyle \int_{x=-2}^{-1} \int_{z = 0}^{-x} \int_{y = 3 x }^{2 z} dy \hspace{4pt} dz \hspace{4pt} dx

Integrating with respect to y y first,

V = ∫ x = − 2 − 1 ∫ z = 0 − x ( 2 z − 3 x ) d z d x V = \displaystyle \int_{x=-2}^{-1} \int_{z = 0}^{-x} (2z - 3 x) \hspace{4pt} dz \hspace{4pt} dx

Now integrating with respect to z z ,

V = ∫ x = − 2 − 1 ( x 2 + 3 x 2 ) d x = ∫ x = − 2 − 1 4 x 2 d x V = \displaystyle \int_{x=-2}^{-1} (x^2 + 3 x^2) \hspace{4pt} dx = \displaystyle \int_{x=-2}^{-1} 4 x^2 \hspace{4pt} dx

Finally, integrating with respect to x x

V = 4 ( 1 3 ) ( ( − 1 ) 3 − ( − 2 ) 3 ) = 4 3 ( 7 ) = 28 3 V = \displaystyle 4 \left( \dfrac{1}{3} \right) ( (-1)^3 - (-2)^3 ) = \dfrac{4}{3} \left( 7 \right) = \dfrac{28}{3}

Therefore, the answer is 28 + 3 = 31 28 + 3 = \boxed{31}

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