Respuesta :
We can define the base of each equilateral section as [tex]b(x) = \sin(x)[/tex] since this is the distance from the x-axis to the function [tex]y=\sin(x)[/tex].
An equilateral triangle's area can have a function defined solely by the base. Since all sides are equal, the height of the triangle can be found via pythagorean theorem as: [tex]A(b)=\frac{1}{2}b\sqrt{b^2+\left (\frac{b}{2} \right)^2} = \frac{b}{2}\sqrt{\frac{5b^2}{4}} [/tex]
But we know that the base [tex]b(x)[/tex] is a function of x. We also can then state that Area [tex]A(b(x))[/tex] is also a function of x. Specifically: [tex]A(x) = \frac{\sin(x)}{2}\sqrt{\frac{5\sin^2(x)}{4}}[/tex].
Then if we integrate the area function A(x) from 0 to pi, we get the total volume:
[tex]\displaystyle \int_0^{\pi} A(x) dx = \int_0^{\pi} \frac{\sin(x)}{2}\sqrt{\frac{5\sin^2(x)}{4}} dx = \int_0^{\pi} \frac{\sqrt{5} \sin^2(x)}{4}dx[/tex]
An equilateral triangle's area can have a function defined solely by the base. Since all sides are equal, the height of the triangle can be found via pythagorean theorem as: [tex]A(b)=\frac{1}{2}b\sqrt{b^2+\left (\frac{b}{2} \right)^2} = \frac{b}{2}\sqrt{\frac{5b^2}{4}} [/tex]
But we know that the base [tex]b(x)[/tex] is a function of x. We also can then state that Area [tex]A(b(x))[/tex] is also a function of x. Specifically: [tex]A(x) = \frac{\sin(x)}{2}\sqrt{\frac{5\sin^2(x)}{4}}[/tex].
Then if we integrate the area function A(x) from 0 to pi, we get the total volume:
[tex]\displaystyle \int_0^{\pi} A(x) dx = \int_0^{\pi} \frac{\sin(x)}{2}\sqrt{\frac{5\sin^2(x)}{4}} dx = \int_0^{\pi} \frac{\sqrt{5} \sin^2(x)}{4}dx[/tex]
