HSC Extension 1 and 2 Mathematics/Integration

Area

• Fundamental Theorem of Calculus: $\underset{a}{\overset{b}{\int }}f(x)dx=F(b)-F(a)$, 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 $V=\pi \underset{a}{\overset{b}{\int }}{y}^{2}dx$

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): $\underset{a}{\overset{b}{\int }}f(x)dx\approx \frac{1}{2}\times \stackrel{=h}{\overbrace{\frac{b-a}{n}}}[f(a)+f(b)]$
• $n$-intervals ($n+1$ function values): $\underset{a}{\overset{b}{\int }}f(x)dx\approx \frac{h}{2}[f(a)+2\sum f(x_i)+f(b)]$

Simpson's rule

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