HSC Extension 1 and 2 Mathematics/Integration

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Area[edit]

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

Area between two curves[edit]

Volume of solids of revolution[edit]

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[edit]

Trapezoidal rule[edit]

  • 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[edit]

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