Find-the-area-of-the-surface-generated-by-revolving-the-curve-x-y-4-4-1-8y-2-about-the-x-axis-given-1-y-2-

Question Number 72789 by Learner-123 last updated on 02/Nov/19

F i n d t h e a r e a o f t h e s u r f a c e g e n e r a t e d

b y r e v o l v i n g t h e c u r v e x = \frac{y^{4}}{4} + \frac{1}{8 y^{2}}

a b o u t t h e x - a x i s . (g i v e n : 1 ⩽ y ⩽ 2)

Commented by MJS last updated on 02/Nov/19

about the x - axis ?

Commented by Learner-123 last updated on 03/Nov/19

y e s, s i r!

I n b o o k t h e q u e s . s a y s a b o u t t h e x - a x i s . .

Commented by MJS last updated on 03/Nov/19

{it}^{'} s not possible about the x - axis

we would need to transform to y = f (x)

this can be done but within the borders

given for y we get

y = 2 \sqrt[4]{\frac{x}{3}} \sqrt{\cos (\frac{π}{6} + \frac{1}{3} \arcsin \frac{3 \sqrt{3}}{32 \sqrt[3]{x^{2}}})}

and we cannot easily go on with this \dots try

if you like; -)

2 π \underset{\frac{3}{8}}{\int^{\frac{129}{32}}} y \sqrt{1 + {(y^{'})}^{2}} d x

Commented by Learner-123 last updated on 03/Nov/19

y e s, i a g r e e, t h a n k s s i r .

Answered by MJS last updated on 02/Nov/19

about the y - axis

the formula is

\underset{a}{\int^{b}} x \sqrt{1 + {(x^{'})}^{2}} d y

x = \frac{2 y^{6} + 1}{8 y^{2}} \Rightarrow x^{'} = \frac{4 y^{6} - 1}{4 y^{3}}

2 π \underset{1}{\int^{2}} \frac{2 y^{6} + 1}{8 y^{2}} \sqrt{1 + {(\frac{4 y^{6} - 1}{4 y^{3}})}^{2}} d y =

= 2 π \underset{1}{\int^{2}} \frac{2 y^{6} + 1}{8 y^{2}} \sqrt{\frac{16 y^{12} + 8 y^{6} + 1}{16 y^{6}}} d y =

= 2 π \underset{1}{\int^{2}} \frac{2 y^{6} + 1}{8 y^{2}} \sqrt{{(\frac{4 y^{6} + 1}{4 y^{3}})}^{2}} d y =

= 2 π \underset{1}{\int^{2}} \frac{2 y^{6} + 1}{8 y^{2}} \times \frac{4 y^{6} + 1}{4 y^{3}} d y =

= \frac{π}{2} \underset{1}{\int^{2}} y^{7} d y + \frac{3 π}{8} \underset{1}{\int^{2}} y d y + \frac{π}{16} \underset{1}{\int^{2}} \frac{d y}{y^{5}} =

= {[\frac{π}{16} y^{8} + \frac{3 π}{16} y^{2} - \frac{π}{64 y^{4}}]}_{1}^{2} = \frac{16911 π}{1024}

Commented by Learner-123 last updated on 03/Nov/19

t h a n k s s i r .