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 © Andrew Bone. All rights reserved. Created : February 5, 2015

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1750 - 1848

Caroline Herschel has traditionally been neglected by history, living in the shadow of her famous brother, William Herschel. But recent research has demonstrated that she made valuable contributions to science in her own right.

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