# Further Maths - Volumes of Revolutions

> Source: https://ollybritton.com/notes/a-level/further-maths/topics/volumes-of-revolutions/ · Updated: 2020-10-08 · Tags: further-maths, safe-to-post-online, integration, school

##### 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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