Question

The volume of a solid revolution generated by rotating the curve y = f(x) and x = f(y) between x = a and x = b , y = a and y = b through 360 degrees about the x-axis and y-axis is given
$V_{x} =\int\limits^b_a {\pi y^{2} } \, dx$ and $V_{y} =\int\limits^b_a {\pi x^{2} } \, dy$

The diagram shows the line y = 1, the line y = 4 and part of the curve $y=3x^2$. The shaded region is rotated through 360 degrees about the y-axis. Find the exact value of the volume of revolution obtained. Leave your answer in pi.

Answer 1

Answer:

$\dfrac{5}{2}\pi$

Step-by-step explanation:

Rotation about the y-axis

$\textsf{Volume}=\displaystyle \int^b_a \pi x^2\:\text{d}y$

where:

• b = upper limit
• a = lower limit
• x is a function of y

Given function of y:  $y = 3x^2$

Rewrite the given function as a function of y:

$\implies x^2=\dfrac{1}{3}y$

Substitute the values into the formula:

$\implies \displaystyle \int^4_1 \dfrac{1}{3}\pi y\:\:\text{d}y$

$\boxed{\begin{minipage}{5 cm}\underline{Terms multiplied by constants}\\\\ \displaystyle \int ay^n\:\text{d}y=a \int y^n \:\text{d}y \end{minipage}}$

$\boxed{\begin{minipage}{4 cm}\underline{Integrating y^n }\\\\ \displaystyle \int y^n\:\text{d}y=\dfrac{y^{n+1}}{n+1}+\text{C} \end{minipage}}$

Take out the constant and integrate:

$\begin{aligned}\implies \dfrac{1}{3}\pi\displaystyle \int^4_1 y\:\:\text{d}y & = \dfrac{1}{3}\pi \left[\dfrac{1}{2}y^2\right]^4_1\\\\& =\dfrac{1}{3}\pi \left[\dfrac{1}{2}(4)^2-\dfrac{1}{2}(1)^2\right]\\\\&=\dfrac{1}{3}\pi\left[8-\dfrac{1}{2}\right]\\\\&=\dfrac{5}{2}\pi \end{aligned}$

Therefore, the exact value of the volume of revolution is:

$\dfrac{5}{2}\pi$

Learn more about integration here:

brainly.com/question/27988986

brainly.com/question/27805589