Question

You will be using the function f(x) = 1/x

1) Describe the shape of a typical cross-sectional slice of the solid and then write down the formula of the area of that cross section.

2) Use an integral to express the total volume of this solid between x=1 and infinity. Explain why the integral represents the volume described.

Answer 1

lets take a randon solid whose cross is eircllar. And surface of solid is rotalingy.= f(x) aberit x-azis section generated ar فكار area of dix any² d. avolume of differential disc . dv - ( 42) height). area dV = TY²da if we add att the differential areas we get volume of whole selid from de, den Jav= yhdx = ( ( f (x) dx this, the integral represents volume of solid of revolution of flow babout x-axis

f() lets take x70 x-axis, la surface @ when we rotate flas about is generated.c the slice (cross sections / will be a circular disc whose rading is from = x volume of having thicknene da radius dV= Iy²da As small and differential cylinders of varying radius will make a solid gratis fav=styrda = on du = lim ASRI da The Synode - lich ca volume of solid = TT units = π -7 lim (1-1) = Rool R

V Eseblaination ich we rotate y = f(x) about & aris we get crow section as circle area of circle = 72 = a(fa) 2 then if we keep if we multiply area by differential length (or pile up duss) vae et differential of we get volume of the disc as it (fonda this disc is actually a cylinder having differential height (da). Now, if we sum op all the such cylinden we end up getting a siled speape what rading is varging as front and height changes from 1 to .. (as in our case) Syz dx = lot de represents Volume of solid object becaus, that sal integral yada actually sums up all the cylinden of de height craving variable radius y = f(x) = 2 .