User:Editeur24/shellintegration

- Shell integration

Consider the ring-cake shape created by rotating the region bounded by the line x = 0, the line $$x = \sqrt{\pi/2}$$, and the curve $$y = \sin (x^2)$$   in the x-y-plane, around the y-axis into the z-dimension. We can divide this shape into circular city-wall shells. The length of each shell is its circumference, which is $$2\pi \cdot radius$$: $$2 \pi x$$ in this context. The width of the shell is dx, so in two dimensions the area is $$2 \pi x dx$$. The height of the shell is $$y = f(x) = \sin (x^2)$$, so the volume of each shell is $$2 \pi x \sin (x^2)dx$$. Adding up all the shells as x changes, we come out with
 * $$ volume = \int_0^{\sqrt{\pi/2}} 2 \pi x \sin (x^2)dx =\int_0^{ \pi/2  } \pi   \sin (u)du =  \Big|_0^{ \pi/2 } - \pi   \cos (u) = \pi ( 0 - -1 ) = \pi,    $$

where this integral has been solved by the method of substitution setting $$u = x^2$$ so $$du = 2x dx$$ and the bounds change from 0 and $$\sqrt{\pi/2}$$ to 0 and $$\pi/2$$.

Keep the old example too,e ven tho it is not as well written. Two examples are good to have. Comment on the Talk page on what I have done, and its deficiencies.