Gabriel\'s horn is formed by revolving the curve y=1/x for x ? [1,infinity) abou

Question

Gabriel's horn is formed by revolving the curve y=1/x for x ? [1,infinity) about the x-axis. Find the volume inside Gabriel's horn. What is the volume?

Explanation / Answer

the first question ---------------------- 0 integral[cos(x)/sqrt(sin(x))] dx pi/2 let sin(x) = t cos(x) = sqrt(1-t^2) dx = dt/sqrt(1-t^2) as x varies as pi/2->0 t varies as 1->0 therefore 0 integral[sqrt(1-t^2)/sqrt(t)] dt/sqrt(1-t^2) 1 0 integral[1/sqrt(t)]dt 1 0 sqrt(t)/1/2 1 0 2sqrt(t) 1 2[0-1] = - 2 the second question ------------------------ volume of a solid of revolution around x axis is b integral[pi * y^2] dx a infinity integral[pi * x^-2] dx 1 infinity -pi * [1/x] 1 -pi * [1/infinity - 1/1] -pi * [0 - 1] pi which is a constant and hence finite the third question ----------------------- It's basically same as the first one except with the limits Here x varies from 0 to pi/2 so, t (which is sin(x)) varies from sin(0) to sin(pi/2) ie t varies from 0 to 1. Hope it helps :)

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