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
Key Terms to Review (17)
Archimedes: Archimedes was an ancient Greek mathematician and physicist known for his work in geometry, calculus, and fluid mechanics. He laid the groundwork for integral calculus through his method of exhaustion.
Area under the curve: The area under the curve is a measure of the region bounded by a given function, the x-axis, and vertical lines at two specified points. It is often computed using definite integrals.
Constant multiple law for limits: The Constant Multiple Law for limits states that the limit of a constant multiplied by a function is equal to the constant multiplied by the limit of the function. Mathematically, if $\lim_{{x \to c}} f(x) = L$, then $\lim_{{x \to c}} [k \cdot f(x)] = k \cdot L$ where $k$ is a constant.
Dummy variable: A dummy variable is a placeholder variable used in integration to represent the variable of integration. It has no impact on the final value of the definite integral.
Index: An index is a numerical or symbolic label used to specify elements within a sequence or sum. In calculus, indices often denote the terms in a sum representing an approximation.
Left-endpoint approximation: Left-endpoint approximation is a method used to estimate the area under a curve on a given interval by summing the areas of rectangles whose heights are determined by the function's value at the left endpoints of subintervals.
Limit: In mathematics, the limit of a function is a fundamental concept that describes the behavior of a function as its input approaches a particular value. It is a crucial notion that underpins the foundations of calculus and serves as a building block for understanding more advanced topics in the field.
Lower sum: A lower sum is an approximation of the area under a curve using the sum of the areas of rectangles that lie entirely below the curve. It is calculated by taking the infimum (or minimum) value of the function within each subinterval.
Method of exhaustion: The method of exhaustion is an ancient mathematical technique used to find the area under a curve by inscribing and circumscribing polygons. It involves calculating the areas of these polygons and taking the limit as the number of sides increases indefinitely.
Partition: A partition of an interval $[a, b]$ is a finite sequence of points that divide the interval into smaller subintervals. These points are used to approximate areas under curves in numerical integration.
Regular partition: A regular partition of an interval $[a, b]$ is a division of this interval into subintervals of equal length. It is used to approximate areas under curves by dividing the area into rectangles or other simple shapes.
Riemann sum: A Riemann sum is a method for approximating the total area under a curve on a graph, otherwise known as an integral. It sums up the areas of multiple rectangles to estimate this area.
Right-endpoint approximation: Right-endpoint approximation is a method to estimate the area under a curve by summing the areas of rectangles. The height of each rectangle is determined by the function value at the right endpoint of each subinterval.
Sigma notation: Sigma notation, also known as summation notation, is a way to represent the sum of a sequence of terms. It uses the Greek letter sigma ($\sum$) to indicate that a series of terms should be added together.
Summation notation: Summation notation is a mathematical symbol used to represent the sum of a sequence of terms. It is denoted by the capital Greek letter sigma ($\sum$) followed by an expression that defines the terms to be added.
Sums and powers of integers: Sums and powers of integers are fundamental concepts in calculus used to approximate areas under curves. These sums often appear as part of Riemann sums and power series.
Upper sum: An upper sum is an approximation of the area under a curve using the sum of areas of rectangles that lie above the curve over each subinterval. The height of each rectangle is determined by the maximum value of the function within that subinterval.