HSC Extension 1 and 2 Mathematics/Integration

Area

 * Fundamental Theorem of Calculus: $$\int_a^b f(x) dx = F(b) - F(a)$$, where $$\frac{d}{dx} F(x) = f(x)$$

Volume of solids of revolution
Recall that the volume of a solid can be found by $$V = Ad\ $$ where $$A$$ is the cross-sectional area and $$d\ $$ is the depth of the solid, which is perpendicular to the cross-sectional area.

Similarly, the volume of solids with circular cross sections can be calculated by
 * rotating a curve about an axis (generally $$x\ $$ or $$y\ $$ axis)
 * integrating to sum the areas of the slices of circles

Since the area of a circle is $$A = \pi r^2\ $$, then the integral to evaluate the volume of a solid generated by revolving it around the $$x$$-axis is $$V = \pi \int_a^b y^2 dx$$

Notice this is a sum of areas of the "slices" of circular cross sections of the solid, i.e. $$\sum \pi r^2 = \pi \sum r^2$$.

Trapezoidal rule

 * One interval (2 function values): $$\int_{a}^{b}f(x)dx \approx \frac{1}{2} \times \overbrace{\frac{b - a}{n}}^{= h} [f(a)+f(b)]$$
 * $$n\ $$-intervals ($$n + 1\ $$ function values): $$\int_{a}^{b}f(x)dx \approx \frac{h}{2}\left[f(a) + 2\sum f(x_i) + f(b)\right]$$

Simpson's rule
$$\int_{a}^{b}f(x)dx \approx \frac{b - a}{6} \left [ f(a) + 4f \left ( \frac{a + b}{2} \right ) + f(b) \right ]$$