Respuesta :
The volume of the solid whose base is a circle centered at the origin with a radius r is 16r³/3 cubed unit.
What is the equation of circle?
The equation of the circle is the equation which is used to represent the circle in the algebraic equation form, with the value of center point in the coordinate plane and measure of radius.
The equation of the circle can be given as,
[tex]x^2+y^2=r^2[/tex]
Here, (r) is the radius of the circle.
Rewrite this equation as,
[tex]y^2=r^2-x^2\\y=\sqrt{r^2-x^2}[/tex]
This will give you the half the length of the side. Thus, the length of the side of the square is,
[tex]s=2y=2\sqrt{r^2-x^2}[/tex]
The area of the square it square of its side. Thus, the area of the square is,
[tex]A(x)=s^2\\A(x)=(2\sqrt{r^2-x^2})^2\\A(x)=4(r^2-x^2})[/tex]
The volume of the solid can be got by integrating the area of the cross-section.
[tex]V=\int\limits {A(x)} \, dx[/tex]
Here, A(x), provides the area of the square at the point x. Thus,
[tex]V=\int\limits {A(x)} \, dx\\V=\int\limits_{-r}^r {4(r^2-x^2)} \, dx\\V=[ {4(r^2x-\dfrac{x^3}{3})}]^r_{-r}\\V= {4(r^2(r)-\dfrac{(r)^3}{3})}- {4(r^2(-r)-\dfrac{(-r)^3}{3})}\\V=4[r^3-\dfrac{r^3}{3}}+ {r^3-\dfrac{r^3}{3}]\\V=\dfrac{16r^3}{3}[/tex]
Thus, the volume of the solid whose base is a circle centered at the origin with a radius r is 16r³/3 cubed unit.
Learn more about the equation of circle here;
https://brainly.com/question/1506955
