Further Maths - Volumes of Revolutions

What volumes of revolution are

In summary, what is volumes of revolutions?

volumes-of-revolution-summary

Finding volumes by integrating.

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

line-rotated-makes-cone

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?

volume-revolution-x-axis-function

\[\int^b _ a \pi f(x)^2 dx\]

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

revolution-as-sum-of-cylinders

Cylinders.

If you have a cylinder with radius $y$ and width $dx$, then what is the formula for the volume of that cylinder?

elemental-cylinder-volume

\[\pi y^2 dx\]

Rotating around each axis

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

volume-revolution-x-axis-y-form

\[\int^b _ a \pi y^2 dx\]

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

volume-revolution-y-axis-y-form

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?

volume-revolution-y-axis-function

\[\int^b _ a \pi f^{-1}(x) dx\]

Compound regions

How would you find the rotated volume of ?

revolution-difference-of-volumes

(a big volume minus a little volume)

How would you find the rotated volume of 2?

revolution-combination-of-curves

(a combination of two curves)

How would you find the rotated volume of 3?

revolution-intersection-of-curves

(the intersection of two curves)

The cone shortcut

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

straight-line-use-cone-formula

Using the formula for the volume of a cone.

Parametric curves

What is the formula for the volume of revolution around the $x$-axis for a parametric curve defined with $x = f(t)$ and $y = g(t)$?

parametric-revolution-x-axis-formula

\[\pi \int^{t = p} _ {t = q} y^2 \frac{\text{d}x}{\text{d}t} dt\]

What is the formula for the volume of revolution around the $y$-axis for a parametric curve defined with $x = f(t)$ and $y = g(t)$?

parametric-revolution-y-axis-formula

\[\pi \int^{t = p} _ {t = q} x^2 \frac{\text{d}y}{\text{d}t} dt\]

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

parametric-change-limits-to-t

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?

parametric-revolution-changes-summary

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?

parametric-y-squared-substitution

\[y^2 \frac{\text{d}x}{\text{d}t}\]

If you normally use $x^2$ when finding the volume of revolution, how would this change for integrating parametrically?

parametric-x-squared-substitution

\[x^2 \frac{\text{d}y}{\text{d}t}\]

Remembering to square

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

always-square-the-function

Square the function!

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

revolution-not-working-square-it

You probably didn’t square the function.

Closed parametric curves

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

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

closed-curve-limits-range-of-motion

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