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Use the general slicing method to find the volume of the following solid. The s...

Feb 8, 2024

Use the general slicing method to find the volume of the following solid. The solid with a semicircular base of radius 13 whose cross sections perpendicular to the base and parallel to the diameter are squares

Generated Graph

Solution by Steps

step 1

To find the volume of the solid, we use the general slicing method, where the cross-sectional area perpendicular to the x-axis is a square

step 2

The side length of each square is equal to the diameter of the semicircle at that x-value, which is

2 \sqrt{13^2 - x^2}

step 3

The area of each square is the side length squared, which is

(2 \sqrt{13^2 - x^2})^2 = 4(13^2 - x^2)

step 4

The volume is found by integrating the area function from

x = 0

x = 13

, which is the radius of the semicircle

step 5

The integral to calculate the volume is

\int_0^{13} 4(13^2 - x^2) \, dx

step 6

Using the Asksia-LL calculator result, the integral evaluates to

\frac{17576}{3}

or approximately

5858.7

Answer

The volume of the solid is

\frac{17576}{3}

cubic units.

Key Concept

Volume by Integration of Cross-Sections

Explanation

The volume of a solid with known cross-sectional area can be found by integrating this area along the axis perpendicular to the cross-sections. In this case, the cross-sections are squares, and their area varies with x as we move along the axis.