A calculus problem by Rahil Sehgal

Calculus Level 3

Consider a cuboid with a opposite corners at ( 0 , 0 , 0 ) (0,0,0) and ( 1 , 2 , 3 ) (1,2,3) .

If this cuboid has integer dimensions and an integer volume, compute its volume.


The answer is 6.

Rahil Sehgal by Rahil Sehgal , 20, India

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1 solution

Using the rectangular coordinate system the volume of the block is given by:

V = ∫ 0 3 ∫ 0 2 ∫ 0 1 d x d y d z = ∫ 0 3 ∫ 0 2 x ∣ 0 1 d y d z = ∫ 0 3 ∫ 0 2 1 d y d z = ∫ 0 3 y ∣ 0 2 d z = ∫ 0 3 2 d z = 2 z ∣ 0 3 = 6 \begin{aligned} V & = \int_0^3 \int_0^2 \int_0^1 dx \ dy \ dz \\ & = \int_0^3 \int_0^2 x \ \bigg|_0^1 \ dy \ dz \\ & = \int_0^3 \int_0^2 1 \ dy \ dz \\ & = \int_0^3 y \ \bigg|_0^2 \ dz \\ & = \int_0^3 2 \ dz \\ & = 2z \ \bigg|_0^3 \\ & = \boxed{6} \end{aligned}

3 Helpful 0 Interesting 0 Brilliant 0 Confused

Wrong. The cuboid could have dimensions 1/2, 1/3, (sqrt491)/6 and a volume of (sqrt491)/36.

Pi Han Goh - 4 years, 1 month ago

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