simplifies complex sums, making it easier to work with series and sequences. It's a powerful tool for representing patterns and calculating totals, setting the stage for more advanced mathematical concepts.
Area estimation using rectangles and Riemann sums are key techniques for approximating definite integrals. These methods provide a visual and practical approach to understanding the fundamental theorem of calculus and its applications.
Sigma Notation and Summations
Summations with sigma notation
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Unit 1: Introduction to integration – National Curriculum (Vocational) Mathematics Level 4 View original
Sigma notation compactly represents the sum of a series of terms using the Greek letter Σ
The of summation (usually i or n) is written below the Σ symbol
The starting and ending values of the index are written above and below the Σ, respectively (∑i=1n)
The general term of the series is written to the right of the Σ (∑i=1nai)
To interpret a given summation, identify the starting and ending values of the index and determine the general term of the series
Evaluate the summation by substituting the index values and simplifying (∑i=13i2=12+22+32=14)
Approximating Areas
Area estimation using rectangles
Approximating the area under a curve using rectangles is a fundamental concept in integral calculus
Divide the interval [a,b] into n subintervals of equal width Δx=nb−a
Construct rectangles on each subinterval with width Δx
The height of each rectangle can be determined by the function value at the left endpoint (left ), right endpoint (right Riemann sum), or midpoint (midpoint Riemann sum) of the subinterval
Calculate the area of each rectangle by multiplying its height by its width (Ai=f(xi∗)Δx)
Sum the areas of all the rectangles to approximate the total (A≈∑i=1nf(xi∗)Δx)
As the number of subintervals n increases, the approximation becomes more accurate (limn→∞∑i=1nf(xi∗)Δx=∫abf(x)dx)
Riemann sums for definite integrals
A Riemann sum is a method for approximating the definite integral of a function f(x) over an interval [a,b]
Divide the interval [a,b] into n subintervals of equal width Δx=nb−a
Choose a sample point xi∗ within each subinterval [xi−1,xi] (left endpoint, right endpoint, or midpoint)
Evaluate the function at each sample point to obtain f(xi∗)
The Riemann sum is given by ∑i=1nf(xi∗)Δx
The definite integral ∫abf(x)dx is the of the Riemann sum as n approaches infinity and Δx approaches zero (∫abf(x)dx=limn→∞∑i=1nf(xi∗)Δx)
Riemann sums provide a way to approximate the value of a definite integral numerically (∫01x2dx≈∑i=1100(100i)2⋅1001)
Approximation techniques and limits
The concept of area under a curve is fundamental to understanding definite integrals
Approximation methods, such as Riemann sums, are used to estimate the area under a curve
A of an interval [a,b] divides it into subintervals, which forms the basis for approximation techniques
The limit process is crucial in refining approximations to obtain exact values of definite integrals