Answer:
[tex]\dfrac{5}{2}\pi[/tex]
Step-by-step explanation:
Rotation about the y-axis
[tex]\textsf{Volume}=\displaystyle \int^b_a \pi x^2\:\text{d}y[/tex]
where:
- b = upper limit
- a = lower limit
- x is a function of y
Given function of y: [tex]y = 3x^2[/tex]
Rewrite the given function as a function of y:
[tex]\implies x^2=\dfrac{1}{3}y[/tex]
Substitute the values into the formula:
[tex]\implies \displaystyle \int^4_1 \dfrac{1}{3}\pi y\:\:\text{d}y[/tex]
[tex]\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}}[/tex]
[tex]\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}}[/tex]
Take out the constant and integrate:
[tex]\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}[/tex]
Therefore, the exact value of the volume of revolution is:
[tex]\dfrac{5}{2}\pi[/tex]
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