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.^[1] 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 $2\pi xdx$ . 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 $du=2xdx$ 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.

^ The $y=\sin(x^{2})$ example is taken from Calculus: Early Transcendentals, 2nd Edition, by William Briggs, Lyle Cochran, Bernard Gillett & Eric Schulz, p. 437.

[1] The $y=\sin(x^{2})$ example is taken from Calculus: Early Transcendentals, 2nd Edition, by William Briggs, Lyle Cochran, Bernard Gillett & Eric Schulz, p. 437.

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