# HSC Extension 1 and 2 Mathematics/Integration

## Area

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

## Volume of solids of revolution

Recall that the volume of a solid can be found by ${\displaystyle V=Ad\ }$ where ${\displaystyle A}$ is the cross-sectional area and ${\displaystyle d\ }$ 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 ${\displaystyle x\ }$ or ${\displaystyle y\ }$ axis)
• integrating to sum the areas of the slices of circles

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

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

## Approximate integration

### Trapezoidal rule

• One interval (2 function values): ${\displaystyle \int _{a}^{b}f(x)dx\approx {\frac {1}{2}}\times \overbrace {\frac {b-a}{n}} ^{=h}[f(a)+f(b)]}$
• ${\displaystyle n\ }$-intervals (${\displaystyle n+1\ }$ function values): ${\displaystyle \int _{a}^{b}f(x)dx\approx {\frac {h}{2}}\left[f(a)+2\sum f(x_{i})+f(b)\right]}$

### Simpson's rule

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