DEFINITE INTEGRALS & NUMERIC INTEGRATION Calculus answers two very important questions. The first, how to find the instantaneous rate of change, we answered with our study of the derivative. We are now ready to answer the second question: how to find the area of irregular regions. Let us start with simple way how to calculate area under the curve that is not well known shape from geometry Very often, we don’t even have a function, but only measurements taken from real life. Point to point. And we have to find area !!!!

Area under the curve. Left-hand Rectangular Approximation Method (LRAM / Ln ) We could estimate the area under the curve by drawing rectangles touching at their left corners. Approximate area: Right-hand Rectangular Approximation Method (RRAM / Rn ) Approximate area:

Midpoint Rectangular Approximation Method (MRAM / Mn ) Approximate area: M4 = With 8 subintervals: Approximate area: M8 = 6.65624

Is there an exact value for area under the curve? Trapezoidal Approximation Method (Tn) Approximate area: T4 Is there an exact value for area under the curve? L4 = 5.75 M4 = 6.625 T4 = 6.75 M8 = 6.65624 R4 = 7.75

Taking the limit as n approaches infinity all methods give the same result: definite integral. The process of finding the sum of the areas of rectangles to approximate area of a region is called Riemann Sums, after Bernhard Riemann, who pioneered the process. SUM BECOMES INTEGRAL

function 𝑓 over interval 𝑎, 𝑏 . ∆𝑥= 𝑏−𝑎 𝑛 𝑥 𝑖 =𝑎+(𝑛−1)∆𝑥 𝐼𝑛 𝑙𝑖𝑚𝑖𝑡 𝑎𝑠 𝑛→∞ 𝑓 𝑥 𝑖 →𝑓 𝑥 ∆𝑥→𝑑𝑥 lim ∆𝑥→0 𝑖=1 𝑛 𝑓 𝑥 𝑖 ∆𝑥= 𝑎 𝑏 𝑓 𝑥 𝑑𝑥=𝐹 𝑏 −𝐹(𝑎) Leibnitz introduced a simpler notation for the definite integral: 𝑏 𝑎 𝑓(𝑥) definite integral of function 𝑓 over interval 𝑎, 𝑏 .

Area under a Curve When the curve is above the ‘x’ axis, the area is the same as the definite integral : Y Area = y= f(x) x = a x= b x © iTutor. 2000-2013. All Rights Reserved

But when the graph line is below the ‘x’ axis, the definite integral is negative. The area is then given by: Area = y= f(x) © iTutor. 2000-2013. All Rights Reserved

And as you can see definite integral is not automatically area; it is rather NET accumulation of the function over interval [a, b]

variable of integration upper limit of integration Integration Symbol integrand variable of integration (dummy variable) lower limit of integration It is called a dummy variable because the answer does not depend on the variable chosen.

AREA IS ALWAYS POSITIVE IN MATH If 𝑦=𝑓 𝑥 is nonnegative and integrable over a closed interval 𝑎, 𝑏 then the area under the curve 𝑦=𝑓 𝑥 from 𝑎 to 𝑏 is the definite integral of 𝑓 from 𝑎 to 𝑏. 𝐴= 𝑎 𝑏 𝑓 𝑥 𝑑𝑥 If 𝑦=𝑓 𝑥 is negative and integrable over a closed interval 𝑎, 𝑏 , then the area under the curve 𝑦=𝑓 𝑥 from 𝒂 to 𝒃 is the OPPOSITE of the definite integral of 𝑓 from 𝑎 to 𝑏. 𝐴=− 𝑎 𝑏 𝑓 𝑥 𝑑𝑥 AREA IS ALWAYS POSITIVE IN MATH If 𝑦=𝑓 𝑥 is both positive and negative on closed interval 𝑎,𝑏 , then 𝑎 𝑏 𝑓 𝑥 𝑑𝑥 will NOT give us the area. In general, 𝑎 𝑏 𝑓 𝑥 𝑑𝑥 does NOT give us the area but rather the NET accumulation over the interval from 𝑥=𝑎 to 𝑥=𝑏. In velocity – time graph, net accumulation of integral is displacement the area (absolute value) is distance !!!!!!

If we are integrating by hand, we must decide if and where the graph crosses the x-axis, then split up our interval, manually making negative regions positive. The following property will help accomplish this: 𝑎 𝑏 𝑓 𝑥 𝑑𝑥= 𝑎 𝑐 𝑓 𝑥 𝑑𝑥+ 𝑐 𝑏 𝑓 𝑥 𝑑𝑥 if the calculator is permitted, then evaluate 𝑎 𝑏 𝑓 𝑥 𝑑𝑥 If you use the absolute value function, you don’t need to find the roots.

Difference between the Value of a Definite Integral and Total Area Find the mean (average) value of the function 𝑓 𝑥 =𝑠𝑖𝑛𝑥 over the interval 0, 2𝜋 Total Area 0 2𝜋 𝑠𝑖𝑛𝑥 𝑑𝑥 = = 0 𝜋 𝑠𝑖𝑛𝑥 𝑑𝑥 − 𝜋 2𝜋 𝑠𝑖𝑛𝑥 𝑑𝑥 Value of the Definite Integral 𝜋 2𝜋 𝜋 =−𝑐𝑜𝑠𝑥 − (−𝑐𝑜𝑠𝑥) 0 2𝜋 𝑠𝑖𝑛𝑥 𝑑𝑥 = 2𝜋 −𝑐𝑜𝑠𝑥 = =− −1 − 1 + 1−(−1) −𝑐𝑜𝑠 2𝜋 − −𝑐𝑜𝑠 0 = =4 − 1 − −1 =

Example 1: The graph of 𝑦=𝑓 𝑥 is shown below. If A1 and A2 are positive numbers that represent the areas of the shaded regions, then shaded area A1 and A2 is: A1 – A2 – 2(– A2 ) = A1 + A2 How else? Simple?