In a Problem i. Suppose p > 0 and R is the region bounded above by V- on the lef

Question

In a Problem i. Suppose p > 0 and R is the region bounded above by V- on the left by-1. ,below by V=0, and (a) Sketch the curves V-n, for p = 0.5, p 1 and p = 2 on the same coordinate axes. (b) For which values of p does R have a finite area? (c) For which values of p does the solid obtained by rotating R about the z-axis have a inite volume? (d) For which values of p does the solid obtained by rotating R about the y-axis have a inite volume? You may use comparison tests for parts (c) and (d), but not for part (b).

Explanation / Answer

I) p=1 red colour graph

p =1/2 . blue colour

p = 2, Green colour

b) IN all the three cases, R does not have a finite area as on right side it extends upto infinity. In other words, y=0 is an asymptote to the curve in all the cases.

c) In all the 3 cases, the solid obtained by rotating around x axis will not have finite volume

d) In all 3 cases, the solid obtained by rotating around y axis also will not have finite volume.

i) The curves have the common x intercept = (1,0)

ii) Asymptotes are y=0 i.e. x axis for x tends to infinity and y =0 for x tends to 0.

iii) When p =1/2 the curve is far from both x axis and y axis, and as p increases, closeness to axes increases.

iv) To find max of each curve

y = lnx/p

Find derivative

y' =0 gives x= infinity or ln x = 1/p

Equate first derivative to 0 to get x=infinity or ln x =1/p

When x tends to infinity, the curve gets minimum.

for maximum x = e1/p

Hence when p =2 maximum at (1.648, 0.389)

When p=1 max at (2.718, 0.607)

When p = 1/2 maximum at (e^2, 0.736) = (7.389,0.736)

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