find-the-volume-of-the-solid-formed-by-rotating-the-area-trapped-by-y-sin-x-and-the-x-axis-around-the-line-y-3-for-0-lt-x-lt-pi-

Facebook Tweet Pin

Question Number 90446 by jagoll last updated on 23/Apr/20

find the volume of the solid formed by rotating the area trapped by y = sin x and the x−axis around the line y=3 for 0<x<π

$$\mathrm{find}\:\mathrm{the}\:\mathrm{volume}\:\mathrm{of}\:\mathrm{the}\:\mathrm{solid} \\ $$$$\mathrm{formed}\:\mathrm{by}\:\mathrm{rotating}\:\mathrm{the}\:\mathrm{area} \\ $$$$\mathrm{trapped}\:\mathrm{by}\:\mathrm{y}\:=\:\mathrm{sin}\:\mathrm{x}\:\mathrm{and}\:\mathrm{the} \\ $$$$\mathrm{x}−\mathrm{axis}\:\mathrm{around}\:\mathrm{the}\:\mathrm{line}\:\mathrm{y}=\mathrm{3} \\ $$$$\mathrm{for}\:\mathrm{0}<\mathrm{x}<\pi\: \\ $$

Commented by john santu last updated on 23/Apr/20

vol = π ∫_0 ^π V(x)^2 −v(x)^2 dx = π∫_0 ^π 3^2 −(3−sin x)^2 dx = π∫_0 ^π 6sin x−sin^2 x dx = π∫_0 ^π (6sin x−(1/2)+(1/2)cos 2x) dx now easy to solve

$${vol}\:=\:\pi\:\underset{\mathrm{0}} {\overset{\pi} {\int}}\:{V}\left({x}\right)^{\mathrm{2}} −{v}\left({x}\right)^{\mathrm{2}} \:{dx}\: \\ $$$$=\:\pi\underset{\mathrm{0}} {\overset{\pi} {\int}}\:\mathrm{3}^{\mathrm{2}} −\left(\mathrm{3}−\mathrm{sin}\:{x}\right)^{\mathrm{2}} \:{dx} \\ $$$$=\:\pi\underset{\mathrm{0}} {\overset{\pi} {\int}}\:\mathrm{6sin}\:{x}−\mathrm{sin}\:^{\mathrm{2}} {x}\:{dx} \\ $$$$=\:\pi\underset{\mathrm{0}} {\overset{\pi} {\int}}\:\left(\mathrm{6sin}\:{x}−\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}}\mathrm{cos}\:\mathrm{2}{x}\right)\:{dx} \\ $$$${now}\:{easy}\:{to}\:{solve} \\ $$

Commented by jagoll last updated on 23/Apr/20

$$\mathrm{thank}\:\mathrm{you}\:\mathrm{sir} \\ $$

Facebook Tweet Pin

find-the-volume-of-the-solid-formed-by-rotating-the-area-trapped-by-y-sin-x-and-the-x-axis-around-the-line-y-3-for-0-lt-x-lt-pi-

Leave a Reply Cancel reply