Question

A solid lies between planes perpendicular to the​ x-axis at xequals=0 and xequals=1212. The​ cross-sections perpendicular to the axis on the interval 0less than or equals≤xless than or equals≤1212 are squares with diagonals that run from the parabola y equals negative 2 StartRoot x EndRooty=−2x to the parabola y equals 2 StartRoot x EndRooty=2x. Find the volume of the solid.

Answer 1

Question:

A solid lies between planes perpendicular to the​ x-axis at x=0 and x=12. The​ cross-sections perpendicular to the axis on the interval 0≤x≤12 are squares with diagonals that run from the parabola y=-2√x to the parabola y=2√x. Find the volume of the solid.

Answer:

576

Step-by-step explanation:

Given:

Length of diagonal square:

$D = 2\sqrt{x} - (-2\sqrt{x})$

$D = 4\sqrt{x}$

Here, the diagonal is the hypotenus of a right angle triangle, with leg S, where the square has a side of length S.

Using Pythagoras theorem:

$S^2 + S^2 = D^2$

$S^2 + S^2 = (4\sqrt{x})^2$

$2S^2 = 16x$

Divide both sides by 2

$S^2 = 8x$

Thus,

Area, A = S² = 8x

Take differential volume, dx =

dV = Axdx

dV = 8xdx

Where limit of solid= 0≤x≤12

Volume of solid, V:

V =∫₀¹² dV

V = 8 ∫₀¹² xdx

V = [4x²]₀¹²

V = 4 (12)²

V = 12 * 144

= 576

Volume of solid = 576