Volume

Calculus Level 4

Find the volume of the region bounded by the hyperbolic cylinders x y = 1 , x y = 9 , x z = 4 , x z = 36 , y z = 25 , y z = 49. xy=1, xy=9, xz=4,xz=36,yz=25,yz=49.

Hint : Let x y = u , x z = v , y z = w xy=u, xz=v,yz=w .


The answer is 64.

Jose Sacramento by Jose Sacramento , 66, Portugal

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1 solution

Otto Bretscher
Feb 29, 2016

Performing the suggested change of variables, with Jacobian ∣ ∂ ( x , y , z ) ∂ ( u , v , w ) ∣ = 1 2 u v w \left|\frac{\partial{(x,y,z)}}{\partial{(u,v,w)}}\right|=\frac{1}{2\sqrt{uvw}} , we find the volume V V of the given solid region W W to be V = ∫ W d z d y d x = ∫ 1 9 ∫ 4 36 ∫ 25 49 1 2 u v w d w d v d u = 64 V=\int_{W}dzdydx=\int_{1}^{9}\int_{4}^{36}\int_{25}^{49}\frac{1}{2\sqrt{uvw}}dwdvdu=\boxed{64}

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