Integration is the technique in calculus of finding the area under a curve by slicing it into infinitesimally thin lines, each having a height of x. The heights are then added up to yield the answer. http://www.math.wisc.edu/~keisler/calc.html There are two ways to do integration, geometrically and symbolically. The geometric way involves cutting up the area by hand as described above, but since one cannot draw such fine lines, there will be a degree of inaccuracy. This method is called the Riemann Sum.http://www.math.wisc.edu/~keisler/calc.html The symbolic method does the same thing as the Riemann Sum, but uses functions and formulas instead.http://www.math.wisc.edu/~keisler/calc.html Integration has many applications across many fields like Physics, Economics and Medicine. http://docs.google.com/viewer?a=v&q=cache:IoM-tIw0cQUJ:faculty.citadel.edu/silver/calculus.pdf+calculus+for+business+and+economics&hl=en&gl=us&pid=bl&srcid=ADGEEShRHQCF8k6a0qI4VXe0U2we17nKOlXkSwvg2IDg-tqCiy0mnAuWHHkbS-Q-KuLuYqGvEXub5vaDuyHqcOIFl5vbnC5trgEhc_yxWeGSgMBCfqtmEAjO1qRftK9eL1bxb01FuxBB&sig=AHIEtbQoLC4r9o0ppfDFr9ocuNrY-Chu-g http://or.amatyc.org/Calculus.htm#RealWorld

These techniques were developed by different mathematicians in the 17th Century. http://www.math.wpi.edu/IQP/BVCalcHist/calc1.html The first one was Cavalieri, around 1635, who used the idea of "indivisibles", rectangles of increasingly thin widths, to find the area of a line. http://www.math.wpi.edu/IQP/BVCalcHist/calc1.html Later, Wallis developed integrals further by evaluating the relationship between different polynomials and curves that are governed by different functions, such as a rectangle and y=k, or a triangle and y=kx. http://www.math.wisc.edu/~keisler/calc.html He described the line in both symbolic and geometric terms. Lastly, Leibniz and Newton independently formulized the relationship between area and the function of a tangent line.http://integrals.wolfram.com/about/history/ They discovered the anti-derivative, the idea that the opposite function of the derivative is an integral. If one takes the derivative of a function and then integrates it, the original function will be left. http://www.math.wpi.edu/IQP/BVCalcHist/calc2.html

Many integrals have pre-solved formulas. http://www.integral-table.com/ Gauss, a German Mathematician, made the first table of integrals. http://integrals.wolfram.com/about/history/A number of the most useful integrals are: http://www.math.com/tables/integrals/tableof.htm

• (integral)x^n dx = x(n+1) / (n+1) + C
• (integral)1/x dx = ln|x| + C
• (integral)e^x dx = ex + C
• (integral)b^x dx = b^x / ln(b) + C
• (integral)ln(x) dx = x ln(x) - x + C
• (integral)sinx dx = -cos x + C
• (integral)cosx dx = sin x + C
• (integral)tanx dx = -ln|cos x| + C

Since integration can be tedious and integrals can be complex, many people use computer programs to integrate, such as the Wolfram Integrator online. http://integrals.wolfram.com/about/faq/#11