Question

The solid whose base is the triangle with vertices ​, ​, and and whose cross sections perpendicular to the base and parallel to the​ y-axis are semicircles.

Answer 1

The volume of the solid is $\frac{a^{3}}{3}$ cubic units.

How to find the volume of a solid by slice integration method

First, we need to determine the coefficients linear function behind the hypotenuse of the triangle by solving the following system of linear equations:

$a\cdot m + b = 0$ (1)

$b = a$ (2)

Where:

• $m$ - Slope
• $b$ - Intercept

The solution of the system is: $m = -1$, $b = a$. Thus, the equation of the line is $y = -x + a$.

The integral expression for the solid volume is described below:

$V = \int\limits_{0}^{a} {A(x)} \, dx$ (3)

Where $A(x)$ is the cross section area function.

If we kwow that $A(x) = \pi\cdot y^{2}$, then the volume of the solid is:

$V = \int\limits^a_0 {(-x+a)^{2}} \, dx$ (4)

$V =\int\limits^a_0 {(x^{2}-2\cdot a\cdot x + a^{2})} \, dx$

$V = \int\limits^a_0 {x^{2}} \, dx -2\cdot a\int\limits^a_0 {x} \, dx + a^{2}\int\limits^a_0 {dx}$

$V = \frac{a^{3}}{3}-a^{3}+a^{3}$

$V = \frac{a^{3}}{3}$

The volume of the solid is $\frac{a^{3}}{3}$ cubic units. $\blacksquare$

Remark

The statement is incomplete and poorly formatted, correct form is shown below:

Find the solid whose base is the triangle with vertices $(0, 0)$, $(0, a)$ and $(a, 0)$ and whose cross sections perpendicular to the base and parallel to the y-axis are semicircles.

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