Integration

Need to know

What you need to know

• **Cartesian Arc Length:** For a curve $y=f(x)$ from $x=a$ to $x=b$, the length is $s = \int_a^b \sqrt{1 + (\frac{dy}{dx})^2} \, dx$.
• **Parametric Arc Length:** For a curve $x=x(t), y=y(t)$ from $t=t_1$ to $t=t_2$, the length is $s = \int_{t_1}^{t_2} \sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2} \, dt$.
• **Surface Area (about x-axis, Cartesian):** $A = \int_a^b 2\pi y \sqrt{1 + (\frac{dy}{dx})^2} \, dx$.
• **Surface Area (about x-axis, Parametric):** $A = \int_{t_1}^{t_2} 2\pi y(t) \sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2} \, dt$.

Explanation

Integration's Power-Ups

1. Identify the integral form, often denoted as I_n.
2. Use integration by parts to express I_n in terms of a lower-order integral, like I_{n-1} or I_{n-2}.
3. Repeatedly apply this 'reduction formula' until you reach a simple base case, like I_0 or I_1, which can be integrated directly.
4. Substitute the value of the base case back up the chain to find the value of the original integral.