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. $\sum \pi r^2 = \pi \sum r^2$.

Approximate integration

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

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