Additional Further Calculus

Reduction formulae

Reduction formulae can be used to evaluate integrals using recursive techniques. Often, they are useful for integrals with a function raised to a power, where integration by parts can then be applied.

Arc lengths

The arc length of a curve is shown below.

The arc length of $y=f(x)$ between $a$ and $b$ is given by:

$$s={\int }_b^a\sqrt{1+{\left(\frac{\text{d}y}{\text{d}x}\right)}^2}\text{d}x$$

Or, alternatively, for a parametric curve where $\dot{x}$ and $\dot{y}$ are the derivatives of $x$ and $y$ with respect to $t$ (between $t=a$ and $t=b$):

$$s={\int }_b^a\sqrt{{\dot{x}}^2+{\dot{y}}^2}\text{d}t={\int }_b^a\sqrt{{\left(\frac{\text{d}x}{\text{d}t}\right)}^2+{\left(\frac{\text{d}y}{\text{d}t}\right)}^2}\text{d}t$$

The formulae for arc length are given in the formulae booklet.

Surface areas of revolution

The surface area of revolution for $y=f(x)$ resolved about the $x$-axis between $a$ and $b$ is given by:

$$S_x=2\pi {\int }_a^{b}y\sqrt{1+{\left(\frac{\text{d}y}{\text{d}x}\right)}^2}\text{d}x$$

Or, alternatively, for a parametric curve in terms of $t$ resolved about the $x$-axis between $t=a$ and $t=b$:

$$S_x=2\pi {\int }_a^{b}y(t)\sqrt{{\left(\frac{\text{d}x}{\text{d}t}\right)}^2+{\left(\frac{\text{d}y}{\text{d}t}\right)}^2}\text{d}t$$

The surface area of revolution for $x=f(y)$ resolved about the $y$-axis between $a$ and $b$ is given by:

$$S_y=2\pi {\int }_a^{b}x\sqrt{1+{\left(\frac{\text{d}x}{\text{d}y}\right)}^2}\text{d}y$$

Or, alternatively, for a parametric curve in terms of $t$ resolved about the $y$-axis between $t=a$ and $t=b$:

$$S_y=2\pi {\int }_a^{b}x(t)\sqrt{{\left(\frac{\text{d}x}{\text{d}t}\right)}^2+{\left(\frac{\text{d}y}{\text{d}t}\right)}^2}\text{d}t$$

The formulae for surface areas of revolution are given in the formulae booklet.