Question

The base of a solid s is the region enclosed by the parabola y = 4 − x2 and the x-axis. Cross-sections perpendicular to the y-axis are squares.

Answer 1

The volume of the solid is 32 cubic units.

How to calculate the volume of solids

Given the following parameters

The length of each cross-section is determined by the horizontal distance (parallel to the x-axis) from one end of the parabola to the other.

Since $y = 4 - x^2$, make "x" the subject of the formula to have:

$x^2 = 4-y\\x = \sqrt{4-y}$

The horizontal distance will be expressed as:

$\sqrt{4-y} - (-\sqrt{4-y} ) = 2\sqrt{4-y}$

Next, is to determine the area of each cross-section.

This is the square of the section's side length and it is expressed as:

$f(y)=(2\sqrt{4-y} )^2\\f(y) = 4(4-y)\\f(y) = 16 - 4y$

The volume of the solid will be expressed as:

$v=\int\limits^4_0 {(16-4y)} dy\\ v = 16y - 2y^2 \\v =[16(4)-2(4)^2] - 0\\v=64-32\\v=32 units^3$

Hence the volume of the solid is 32 cubic units.

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