Find the volume of the region bounded by

Question

Find the volume of the region bounded by z=x^2, 0 ≤ x ≤ 4, and the planes y=0, y=6, and z=0.

volume?

Answer 1

Given:

f( x, y ) = x²

Bounded, x: [ 0, 4 ]

Between planes: y = 0, y = 6, z = 0

Find:

The volume bounded the function f by the constraints above.

Solution:

Step 1: Set-up Double Integral to evaluate Volume

V = ∫∫ f( x, y ) dA

Due to being in Cartesian coordinates:

dA = dy dx

V = ∫∫ f( x, y ) dy dx

V = ∫∫ x² dy dx

Step 2: Determine the bounds of integration

The x-bounds are simple since the problem prompt gave them for you.

V = ∫₀⁴∫ x² dy dx

The y-bounds are sort-of simple but are interpreted different than the straight-forward "x can be in this range of values"

The planes y = 0 and y = 6 let you know that the y domain for the integral can only be from 0 to 6 and thus the y-bounds are from 0 to 6

V = ∫₀⁴ ∫₀⁶ x² dy dx

Step 3: Begin evaluating the double-integral

Because x² does not depend on y, integration of the constant 1 w.r.t. y will follow through unhindered.

V = y|₀⁶ * ∫₀⁴ x² dx = ( 6 - 0 ) * ∫₀⁴ x² dx V = 6 * ∫₀⁴ x² dx

The anti-derivative of x² is 1/3 x³

V = 6/3 * x³|₀⁴

V = 2 * ( 4³ - 0³ ) = 2 * 64

∴ V = 128 units³

I hope this helps! Message me in the comments if you have any questions, comments, or concerns on the process or anywhere in-between!