The volume of the solid is
cubic units.
<h3>
How to find the volume of a solid by slice integration method</h3>
First, we need to determine the coefficients <em>linear</em> function behind the hypotenuse of the triangle by solving the following system of linear equations:
(1)
(2)
Where:
- Slope
- Intercept
The solution of the system is:
,
. Thus, the equation of the line is
.
The integral expression for the solid volume is described below:
(3)
Where
is the cross section area function.
If we kwow that
, then the volume of the solid is:
(4)
![V =\int\limits^a_0 {(x^{2}-2\cdot a\cdot x + a^{2})} \, dx](https://tex.z-dn.net/?f=V%20%3D%5Cint%5Climits%5Ea_0%20%7B%28x%5E%7B2%7D-2%5Ccdot%20a%5Ccdot%20x%20%2B%20a%5E%7B2%7D%29%7D%20%5C%2C%20dx)
![V = \int\limits^a_0 {x^{2}} \, dx -2\cdot a\int\limits^a_0 {x} \, dx + a^{2}\int\limits^a_0 {dx}](https://tex.z-dn.net/?f=V%20%3D%20%5Cint%5Climits%5Ea_0%20%7Bx%5E%7B2%7D%7D%20%5C%2C%20dx%20-2%5Ccdot%20a%5Cint%5Climits%5Ea_0%20%7Bx%7D%20%5C%2C%20dx%20%2B%20a%5E%7B2%7D%5Cint%5Climits%5Ea_0%20%7Bdx%7D)
![V = \frac{a^{3}}{3}-a^{3}+a^{3}](https://tex.z-dn.net/?f=V%20%3D%20%5Cfrac%7Ba%5E%7B3%7D%7D%7B3%7D-a%5E%7B3%7D%2Ba%5E%7B3%7D)
![V = \frac{a^{3}}{3}](https://tex.z-dn.net/?f=V%20%3D%20%5Cfrac%7Ba%5E%7B3%7D%7D%7B3%7D)
The volume of the solid is
cubic units. ![\blacksquare](https://tex.z-dn.net/?f=%5Cblacksquare)
<h3>Remark</h3>
The statement is incomplete and poorly formatted, correct form is shown below:
<em>Find the solid whose base is the triangle with vertices </em>
<em>, </em>
<em> and </em>
<em> and whose cross sections perpendicular to the base and parallel to the y-axis are semicircles. </em>
To learn more on volumes, we kindly invite to check this verified question: brainly.com/question/1578538