# Volumes of Revolution Here, a line is wrapped around an arbitrary axis, creating a curve. This means that we can get the formula for its volume: $V = \int_{a}^b \pi y^2 \, dx $ So make sure you substitute the $y$ for any value $x$! If this integral is wrapped around the $y$ axis, we instead use the formula: $V = \int_c^d \pi x^2 \, dy $ To denote the volume. >[!Abstract]- Graphical Volumes of Revolution Intuition # Finding the Arclength We're going to be moving to the James Stewart book now! How delighting... As a beginner exercise, let's take a look at a shape inscribed into a circle, like a hexagon. Finding the "arclength" just becomes a matter of adding up all the side lengths - simple as that! $L = \sum_{i=1}^n |x_{i}|$ Denoting the magnitude of the lengths on the plane. We can monitor the start and end of each line with the points $P(x_i, y_i)$. Now let's extend this definition to an improper curve, such as a sinusoidal. Taking the arclength will require a near-infinite number of linear lengths - giving us an infinite sum: $L = \lim_{n \to \infty } \sum_{i=1}^n |P_{i-1}P_{i}|$ Where the vector we take the sum of denotes each line, giving us a total summation. # Optimisation >[!Success]- Further Generalisation of the Arclength into $x$ and $y$ Components >Note that we can break the points $p$ and $p_{i-1}$ into their constituent $x$ and $y$ coordinates. > >![[Pasted image 20230822142230.png]]