# Approximating Volume & Area Using Integration Many simple volume functions are derived from the methods listed below - i.e. The formula for the volume of a cone ($y= \frac{1}{3} a^2 h$) looks very similar to the power rule. ## Approximating Area - [ ] Net Signed Area vs Total Area $\int^{b}_{a} \vert f(x) \vert dx = \lim_{n \to \infty} \sum^{n}_{i=1} \vert f(x^*_i) \vert \Delta x$ ### The Area Between Curves ## Approximating Volume ![[Pasted image 20220725005619.png|400]] ### The Disk Method ![[Pasted image 20220725011823.png|600]] $V = \int^{b}_{a} \pi \cdot [f(x)^2]dx$ The $[a,b]$ interval is the same when solving for the function in terms of $x$ or $y$. The only difference is replacing $f(x)$ with $f(y)$ in the **integrand**. The interval is changed to be $[f(a), f(b)]$ if the axis is changed. ### The Washer Method ![[Pasted image 20220725012025.png|400]] The washer method is essentially subtracting two **disks** to find the volume of a **washer** $V = \int^{b}_{a} \pi \cdot [f(x)^2 - g(x)^2] dx$ When determining the order of the functions, use the original graph. The mirrored graph (what the projection will be after rotated around the specified axis) is simply there to help depict the **solid of revolution**. ### The Shell Method ![[Pasted image 20220725011122.png|400]] > [!rule] The Shell Method > > $ V = \int^{b}_{a} 2 \pi x \cdot f(x) dx$ > - The standard form of this method rotates the area around the $y$-axis.