Question

A solid lies between planes perpendicular to the​ x-axis at x0 and x. The​ cross-sections perpendicular to the axis on the interval 0x are squares with diagonals that run from the parabola to the parabola. Find the volume of the solid.

Answer 1

The volume of the solid is 900 cubic unit given that the solid lies between planes perpendicular to the x-axis at x = 0 and x = 19, the cross sections perpendicular to the x-axis on the interval 0 ≤ x ≤ 15 are squares with diagonals that run from the parabola y = - 2√x to the parabola y = 2√x. This can be obtained by finding the area of the square using the length of the diagonal.

 

What is the volume of the solid?

Given that, diagonals that run from the parabola y = - 2√x to the parabola   y = 2√x

• The length of the diagonal,

D = 2√x - (-2√x)

D = 4√x

• Using Pythagoras theorem,

D² = s² +  s², where s is the side of the square

(4√x)² = 2s²

16x = 2s²

s² = 8x

s² is the area

• Area A = 8x

Thus,

• volume V = ∫A dx, 0 ≤ x ≤ 15

V = $\int\limits^{15}_0 {8x} \, dx$

V =  $8\int\limits^{15}_0 {x} \, dx$

V = 4 (15² - 0)

V = 4×225

⇒ V = 900 cubic unit

Hence the volume of the solid is 900 cubic unit given that the solid lies between planes perpendicular to the x-axis at x = 0 and x = 19, the cross sections perpendicular to the x-axis on the interval 0 ≤ x ≤ 15 are squares with diagonals that run from the parabola y = - 2√x to the parabola y = 2√x.

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Question: A solid lies between planes perpendicular to the x-axis at x = 0 and x = 19. The cross sections perpendicular to the x-axis on the interval         0 ≤x ≤ 15 are squares with diagonals that run from the parabola y = - 2√x to the parabola y = 2√x. Find the volume of the solid.