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My name is Ann [436]
2 years ago
5

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.

Mathematics
1 answer:
Zanzabum2 years ago
6 0

Answer:

\dfrac{5}{2}\pi

Step-by-step explanation:

<u>Rotation about the y-axis</u>

\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 <u>exact value</u> of the volume of revolution is:

\dfrac{5}{2}\pi

Learn more about integration here:

brainly.com/question/27988986

brainly.com/question/27805589

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