# Geometric integration

## Area under a curve

The integral of a function \$f(x)\$, \$∫_{0}^{b}f(x)dx \$, gives the area enclosed between the curve and the \$x\$-axis, between the \$y\$-axis and \$x=b\$.

Similarly, the integral \$∫_{a}^{b}f(x)dx\$, returns the area between the curve and the \$x\$-axis, between \$x=a\$ and \$x=b\$.

### Fundamental Theorem of Calculus

If \$f\$ is continuous in \$[a,b]\$ and if \$F\$ is any anti-derivative of \$f\$ on \$[a,b]\$, then \$∫_a^{b}f(x)dx=F(b) - F(a)\$

This formulation was set down by Augustin-Lewis Cauchy, 1789-1857, a French mathematician, and in so doing he finally and formally united the two branches of calculus, that of Newton, and that of Leibniz.

## Area between two curves

If \$y_1\$ and \$y_2\$ are continuous on \$a ≤ x ≤ b\$ for all x in \$a ≤ x ≤ b\$, then the area between \$y_1\$ and \$y_2\$ from x = a to x = b is given by:
\$\$∫↙{a}↖{b} (y_1 - y_2)dx \$\$

The method of approximating the area under a curve by the sum of an infinite series of rectangles is known as Riemann sums, after the famous German mathematician, Georg Friedrich Bernhard Riemann, 1826 - 1866.

What is the area of the region bounded by the curves: \$y = x^2\$ and \$y = 1 - x^2\$?

First, find the points of intersection by equating the two equations:

\$x^2=1-x^2\$, ⇒ \$x=±1/{√2}\$

\$A = ∫_{-1/{√2}}^{+1/{√2}} [f(x) - g(x)]dx = ∫_{-1/{√2}}^{+1/{√2}} [1-x^2 -x^2]dx = ∫_{-1/{√2}}^{+1/{√2}} [1-2x^2]dx = [x - 2/{3}x^3]_{-1/{√2}}^{+1/{√2}}\$

\$= 2/{3√2} - (-2/{3√2}) = 0.943\$

## Volume of Revolution

The volume of a solid formed by rotating the area between a function and the x-axis through 360° is:

\$\$∫_a^bπy^2dx\$\$

If a two-dimensional shape is rotated through 360°, a solid, 3D shape results. Integration allows us to calculate the volume of the solid formed by rotation.

Integration: rotation of rectangle through 360° results in a cylinder

A rectangle of height y and thickness dx rotated through 360° forms a disk, or narrow cylinder. This disk has a volume equal to the area of the circle it forms times the disk height (dx):

\$\$V_{disk} = πr^2h = πy^2 dx\$\$

The volume of a 3D shape formed by a series of disks, from a to b on a curve, is:

\$\$∫↙{a}↖{b} πy^2dx\$\$

### Rotation of the area between two curves

Where \$f(x) ≥ g(x)\$, for all \$x\$ in the interval \$[a, b]\$, the volume of rotation formed by rotating the area between the two curves \$2π\$ radians about the \$x\$-axis in the interval \$[a, b]\$ is given by:

\$\$V = π∫_a^b (f(x))^2 - π∫_a^b (g(x))^2dx\$\$ \$\$= π∫_a^b [f(x)^2 - g(x)^2]dx\$\$

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### Quote of the day...

"What's wrong with that?!" shrugged Pau. "Executioners work in Diagony Alley, civil servants in Memo Circle, legislators in Backhand Avenue, lawyers in Waffle Park, doctors in Or Topsy Deadend, marriage agents in Pickadilly Circus - why shouldn't bakers mix in Pudding Lane?"