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 $x=b$.

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.

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:

$x^2=1-x^2$, ⇒ $x=±1/{√2}$

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

$= 2/{3√2} - (-2/{3√2}) = 0.943$

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.

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:

$$∫↙{a}↖{b} πy^2dx$$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:

$$V = π∫_a^b (f(x))^2 - π∫_a^b (g(x))^2dx$$ $$= π∫_a^b [f(x)^2 - g(x)^2]dx$$Content © Renewable-Media.com. All rights reserved. Created : February 5, 2015

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Mathematics is the most important tool of science. The quest to understand the world and the universe using mathematics is as old as civilisation, and has led to the science and technology of today. Learn about the techniques and history of mathematics on ScienceLibrary.info.

1635 - 1703

Robert Hooke was an English polymath and all-round genius, active in an extraordinary range of fields during the English Enlightenment. History has not been fair to Hooke, obscuring his contributions in the shadow of his gigantuan rival, Isaac Newton.

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