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
[tex]V_{x} =\int\limits^b_a {\pi y^{2} } \, dx[/tex] and [tex]V_{y} =\int\limits^b_a {\pi x^{2} } \, dy[/tex]
The diagram shows the line y = 1, the line y = 4 and part of the curve [tex]y=3x^2[/tex]. 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.
Answers
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 limita = lower limitx is a function of yGiven 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]
Learn more about integration here:
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Answers
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Step-by-step explanation:
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