Question

Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line. x

Answer 1

Answer:

$\frac{784}{15} \pi$

Step-by-step explanation:

According to the given situation, the calculation of volume of the solid is shown below:-

Here we will consider the curves that is

$x = 7y^2, x = 7$

Now, rotating the line for the line x which is equals to 7

$7y^2 = 7\\\\y^2 = 1\\\\ y = \pm1$

So, the inner radio is

7 - 7 = 0

and the outer radius is

$7y^2 - 7\\\\ = 7(y^2 - 1)$

Now, the area of cross section is

$A(y) = \pi(7(y^2 - 1))^2\\\\ = 49\pi(y^4 - 2y^2 + 1)$

The volume is

$V = \int\limits^1_{-1} A(y)dy$

now we will put the values into the above formula

$= \int\limits^1_{-1} 49\pi(y^4 - 2y^2 + 1)dy\\\\ = 49\pi(\frac{y^5}{5} - \frac{2y^3}{3} + y)^{-1}\\\\ = 49\pi(\frac{1}{5} - \frac{2}{3} + 1 + \frac{1}{5} - \frac{2}{3} + 1)\\\\ = 49\pi(2 + \frac{2}{5} - \frac{4}{3} )\\\\ = 49\pi(\frac{30+6-20}{15} )\\\\ = \frac{49\pi}{15} (16)$

After solving the above equation we will get

$= \frac{784}{15} \pi$