Further Maths - Volumes of Revolutions

Created: October 08, 2020

In summary, what is volumes of revolutions?

Finding volumes by integrating.

Rotating the line $y = x$ around the $x$ axis creates what shape?

A cone.

If you have a function $f(x)$ and an interval $[a,b]$, how can you find the volume of revolution around the $x$-axis?
\[\int^b_a \pi f(x)^2 dx\]

If normal integration is an infinite summation of rectangles, then volumes of revolutions is an infinite summation of?

Cylinders.

If you have a cylinder with radius $y$ and width $dx$, then what is the formula for the volume of that cylinder?
\[\pi y^2 dx\]

If you have a function $y = …$ on the interval $[a,b]$, how can you find the volume of revolution around the $x$-axis?
\[\int^b_a \pi y^2 dx\]

If you have a function $y = …$ on the interval $[a,b]$, how can you find the volume of revolution around the $y$-axis?

Rearrange for $x = …$

\[\int^b_a \pi x^2 dx\]

If you have a function $f(x)$ and an interval $[a,b]$, how can you find the volume of revolution around the $y$-axis?
\[\int^b_a \pi f^{-1}(x) dx\]

How would you find the rotated volume of PHOTO?

PHOTO (a big volume minus a little volume)

How would you find the rotated volume of PHOTO 2?

PHOTO (a combination of two curves)

How would you find the rotated volume of PHOTO 3?

PHOTO (the intersection of two curves)

2021-10-05

What’s easier than using volumes of revolution for a straight line like $y = 2x + 18$ or $2x + 3y - 5 = 0$?

Using the formula for the volume of a cone.

2021-12-15

What is the formula for the volume of revolution around the $x$-axis for a parametric curve defined with $x = f(t)$ and $y = g(t)$?
\[\pi \int^{t = p}_{t = q} y^2 \frac{\text{d}x}{\text{d}t} dt\]

What is the formula for the volume of revolution around the $y$-axis for a parametric curve defined with $x = f(t)$ and $y = g(t)$?
\[\pi \int^{t = p}_{t = q} x^2 \frac{\text{d}y}{\text{d}t} dt\]

What must you remember to do when integrating a parametric curve?

Change the limits so they are for $t$.

How could you summarise the change you make to a volumes of revolution formula for integrating parametric curve?
  • Change limits
  • Multiply $x^2$ or $y^2$ by the derivative of the other variable with respect to $t$.

If you normally use $y^2$ when finding the volume of revolution, how would this change for integrating parametrically?
\[y^2 \frac{\text{d}x}{\text{d}t}\]

If you normally use $x^2$ when finding the volume of revolution, how would this change for integrating parametrically?
\[x^2 \frac{\text{d}y}{\text{d}t}\]

2022-04-18

What do you always forget on volumes of revolution questions!!!?

Square the function!

“Hmmmm. Why isn’t this volumes of revolutions question working”?

You probably didn’t square the function.

2022-05-30

\[x = \cos\theta + \frac{1}{2}\sin 2\theta\] \[y = -(1 + \cos\theta)\]

How could you tell that the $y$ limits for doing this parametric volumes of revolutions question for a closed curve was $0$ and $-2$?

It’s the “range of motion” for the $y$-component.



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