Answer to Question #187010 in Calculus for Jethro

Question

Use a triple Integral to find the volume of the pyramid P whose base is the square with vertices (1,0,0), (0,1,0), (-1,0,0), and (0,-1,0) and whose top vertex is (0,0,1).

Answer 1

First of all the limits of x, y and z have to be determined.

On seeing the coordinates and their respective quadrants we have,

Limit of z : 0 to -x-y+1

Limit of y : 0 to 1-x

Limit of x : 0 to 1

Volume = 4 * ∫ ₀¹ ∫₀^1-x ∫₀^-x-y+1 dz dy dx

Volume = 4 * ∫ ₀¹ ∫₀^1-x ( - x - y + 1 ) dy dx

Volume = 4 * ∫ ₀¹ [ -x(1 - x) - (1 - x)² /2 + 1-x ] dx

Volume = 4 * ∫ ₀¹ [ -x + x² - 1/ 2 - x²/2 + x +1 - x ] dx

Volume = 4 * ∫ ₀¹ [ x²/2 - x + 1/2 ] dx

Volume = 4 * [ x³ / 6 - x² /2 + x/2 ] ₀¹

Volume = 4 * [ 1/ 6 - 1/2 + 1/2]

Volume = 4/6

Volume = 0.667 cubic units