Find the area of the surface obtained by rotating the curve `y= \sqrt(3 x)` from `x=0` to `x = 4` about the `x`-axis. Answer in square units.

Expert Answers

An illustration of the letter 'A' in a speech bubbles

Hello!

There is the formula for the area of a surface of revolution: if a function `f(x) ` is considered for `x ` from `a ` to `b ` and its graph is rotated over the x-axis, then the surface area is

`A = int_a^b ( 2 pi f ( x ) sqrt ( 1 + ( f ' ( x ) )^2 ) ) dx .`

In our question `a = 0 , ` `b = 4 , ` `f ( x ) = sqrt ( 3 x ) , ` `f ' ( x ) = sqrt ( 3 ) / ( 2sqrt ( x ) ) .`

This way, the area is

`A = int_0^4 ( 2 pi sqrt ( 3 x ) sqrt ( 1 + 3 / (4x) ) ) dx ​= int_0^4 ( pi sqrt(3) sqrt ( 4 x + 3 ) ) dx .`

To make it even simpler, substitute `u = 4 x + 3 , ` `du = 4 dx , ` `u ` from `3 ` to `19 . ` Then we obtain

`A = int_3^19 ( pi sqrt(3)/4 sqrt ( u ) ) du = pi sqrt(3)/4 ( 2 / 3 ) ( u^( 3 / 2 ) )_( u = 3 )^19 = `

`= pi / (2sqrt(3)) ( 19^( 3 / 2 ) - 3^(3/2) ) = pi/6 (19 sqrt(57) - 9).`

It is approximately 70.4 square units.

Approved by eNotes Editorial Team