 # 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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