Question

Refer to the figure and find the volume V generated by rotating the given region about the specified line.
ℛ3 about OC

Answer 1

Volume generated by the areas R2 + R3 around the line OC
= (1/3) pi 1^2 * 3  =  pi sq units

Volume generated by area R2 around OC  is found as follows

the  equation of the curve can be written as  x = y^4 / 81

Integral between limits y =  0 and 3  of  the curve is  

pi INT  x^2 dy    = INT pi  y^8 / 81^2  dx   

= pi [3^9 / 9*81^2] 

= pi/3

So the volume generated by R3 about OC = pi - pi/3  = 2pi/3 

Answer 2

The volume is 69.675.

Integration

It is the reverse of derivation. It is similar to the way of adding the slice to make it whole.

Volume formula

$\rm Volume = \int\limits^a_b {\pi y^{2} } \, dx$

To find

The volume of the region $\rm R_{3}$.

How to find the volume?

The limit is from 0 to 1.

The first function is parabola second is the line.

Then volume will be given by

$\rm Volume = \int\limits^3_0 {\pi y_{1}^{2} } \, dx -\int\limits^3_0 {y_{2}^{2} } \, dx \\\\\rm Volume = \int\limits^3_0 {\pi (3x^{\frac{1}{4} })^{2} } \, dx -\int\limits^3_0 {\pi x^{2} } \, dx \\\\ Volume = 9\pi \int\limits^3_0 {x^{0.5 } \, dx - \pi \int\limits^3_0 {x^{2} } \, dx \\\\Volume = 9\pi [\dfrac{x^{1.5} }{1.5} ]_{0}^{3} - \pi [\dfrac{x^{3} }{3} ]_{0}^{3}\\\\\rm Volume =\dfrac{ 9\pi}{1.5} (3^{1.5} -0^{1.5} ) - \dfrac{\pi }{3} (3^{3} -0^{3} )$

$\rm Volume = 97.945 - 28.27\\\\ \rm Volume = 69.675$

Thus the volume is 69.675.

More about the integration link is given below.

https://brainly.com/question/18651211