Approximating Integrals Using Riemann Sums

Unit: Integration & Accumulation of Change

Chapter: Approximation Integrals using Riemann Sums

Reference: – Riemann sums, Partition of an interval, Left & Right Riemann sum, Midpoint, Lower & Upper Riemann sums, Rectangular & Trapezoidal approximations, Composite Riemann sums, Width & Subintervals, Error Estimation.

 

After studying this chapter, you should be able to:

  • Introduction to Riemann sums & Left, Right & Midpoint sums.
  • Notation & Approximation area.
  • Rectangular Approximation method.
  • Left endpoint rule & Midpoint rule

 

Introduction to Riemann Sums

 

Riemann sums are a calculus method used to approximate the area under a curve by dividing the interval into smaller subintervals and summing the areas of corresponding rectangles or trapezoids. They provide an approximation of the definite integral of a function.

Here are the key aspects of Riemann sums:

  1. Partitioning the Interval: To apply Riemann sums, the interval over which the function is integrated is divided into smaller subintervals. This is done by selecting a partition, which is a set of points that divide the interval into subintervals. The partition is typically denoted by P and can be represented as P = {x0, x1, x2, …, xn}, where x0 is the left endpoint of the interval and xn is the right endpoint.
  2. Subintervals: Each subinterval is determined by consecutive points in the partition. The subintervals have a width determined by the difference between two consecutive points. For example, the first subinterval is [x0, x1], the second is [x1, x2], and so on.
  3. Sample Points: Riemann sums require the selection of sample points within each subinterval. The sample points are used to determine the height of the rectangles or trapezoids. Commonly used methods for selecting sample points include the left endpoint, right endpoint, midpoint, or any arbitrary point within the subinterval.
  4. Types of Riemann Sums: The choice of sample points and the shape of the rectangles or trapezoids determine the type of Riemann sum. There are four commonly used types:
  • Left Riemann Sum: The left endpoint of each subinterval is used as the sample point. The area of each rectangle is determined by multiplying the function value at the left endpoint by the width of the subinterval.
  • Right Riemann Sum: The right endpoint of each subinterval is used as the sample point. The area of each rectangle is determined by multiplying the function value at the right endpoint by the width of the subinterval.
  • Midpoint Riemann Sum: The midpoint of each subinterval is used as the sample point. The area of each rectangle is determined by multiplying the function value at the midpoint by the width of the subinterval.
  • Trapezoidal Riemann Sum: The sample points are chosen as the endpoints of each subinterval. The area of each trapezoid is determined by calculating the average of the function values at the endpoints and multiplying it by the width of the subinterval.
  1. Approximating the Definite Integral: The Riemann sum is obtained by summing the areas of the rectangles or trapezoids corresponding to each subinterval. As the number of subintervals increases, the Riemann sum becomes a better approximation of the definite integral of the function over the given interval.
  2. Notation: Riemann sums are commonly denoted using sigma notation (∑). The general form of the Riemann sum for a function f(x) on the interval [a, b] is:

R = ∑[f(xi*)Δxi]

where xi* represents the sample point within each subinterval, and Δxi is the width of the subinterval.

Left, Right & Mid-point Riemann Sums:

 

  1. Left Riemann Sum: In a left Riemann sum, the left endpoint of each subinterval is used to determine the height of the rectangle. The width of each subinterval is determined by the partitioning of the interval of integration. The left Riemann sum is calculated by multiplying the height of each rectangle by its width and summing up all the resulting areas.
  2. Right Riemann Sum: In a right Riemann sum, the right endpoint of each subinterval is used to determine the height of the rectangle. Similar to the left Riemann sum, the width of each subinterval is determined by the partitioning. The right Riemann sum is calculated by multiplying the height of each rectangle by its width and summing up all the resulting areas.
  3. Midpoint Riemann Sum: In a midpoint Riemann sum, the midpoint of each subinterval is used to determine the height of the rectangle. The width of each subinterval is still determined by the partitioning. The midpoint Riemann sum is calculated by multiplying the height of each rectangle by its width and summing up all the resulting areas.

The choice of using left, right, or midpoint endpoints in Riemann sums affects the accuracy of the approximation. The left and right Riemann sums provide lower and upper bounds on the actual area under the curve, respectively. The midpoint Riemann sum, on the other hand, generally provides a more accurate approximation compared to the left and right sums.

 

Partitioning the Interval & Riemann sum Notation: –

 

Partitioning the Interval:

  1. An interval is a range of values on the number line, often denoted as [a, b], where "a" represents the lower limit and "b" represents the upper limit.
  2. Partitioning the interval involves dividing it into smaller subintervals or intervals.
  3. The process of partitioning determines the width of each subinterval, which affects the accuracy of the Riemann sum approximation.
  4. The partitioning can be done by choosing specific points within the interval to create the subintervals.

Riemann Sum Notation:

  1. Riemann sums are typically represented using sigma (Σ) notation, which is a shorthand way of expressing a sum.
  2. The sigma symbol Σ is followed by the variable that represents the index of summation. This variable is often denoted as "i" or "k".
  3. The lower and upper limits of summation are written as subscripts below and above the sigma symbol, respectively. These limits define the range of values for the index variable.
  4. The expression inside the sigma symbol represents the function evaluated at specific points.
  5. Riemann sum notation combines the function evaluation with the width of the subintervals to calculate the areas of the corresponding rectangles.
  6. Depending on the type of Riemann sum, the expression inside the sigma symbol may involve using left, right, or midpoint endpoints of the subintervals to determine the height of the rectangles.

To put it all together, when working with Riemann sums:

  1. Partition the interval by dividing it into smaller subintervals.
  2. Determine the width of each subinterval.
  3. Choose the appropriate Riemann sum type (left, right, or midpoint).
  4. Use Riemann sum notation with sigma notation to express the approximation.
  5. Calculate the height of each rectangle using the chosen Riemann sum method.
  6. Multiply the height of each rectangle by its width to obtain the area.
  7. Sum up all the areas of the rectangles to approximate the total area under the curve.

 

Approximation Area & Rectangular Approximation Method:

Approximation of Area:

 

  1. In calculus, the problem of finding the exact area under a curve can be challenging. To tackle this, approximation methods are used to estimate the area.
  2. The idea is to divide the region under the curve into smaller, simpler shapes like rectangles and then calculate the sum of their areas.
  3. The sum of these areas provides an approximation of the total area under the curve.
  4. By using more narrower rectangles, the approximation becomes more accurate.

Rectangular Approximation Method:

 

  • Rectangular approximation is one of the simplest methods for estimating the area under a curve.
  • The region under the curve is divided into a series of rectangles, and the sum of their areas is calculated.
  • The width of each rectangle is determined by partitioning the interval of integration.
  • The height of each rectangle is determined based on the function being integrated.
  • The choice of endpoints (left, right, or midpoint) for determining the height of the rectangles depends on the specific Riemann sum being used.
  • The left endpoint rule (or lower sum) uses the left endpoint of each subinterval to determine the rectangle height.
  • The right endpoint rule (or upper sum) uses the right endpoint of each subinterval to determine the rectangle height.
  • The midpoint rule uses the midpoint of each subinterval to determine the rectangle height.
  • Once the heights and widths of the rectangles are determined, the areas of each rectangle are calculated by multiplying the height by the width.
  • The areas of all the rectangles are then summed to obtain an approximation of the area under the curve.

Example: – Approximate the integral of the function f(x) = x2 on the interval [0, 4] using four subintervals and the midpoint Riemann sum.

 

Solution:

Step 1: Determine the width of each subinterval.

The interval [0, 4] is divided into four equal subintervals, so the width of each subinterval is (4-0)/4 = 1.

 

Step 2: Calculate the midpoints of the subintervals.

The midpoints of the subintervals are 0.5, 1.5, 2.5, and 3.5.

 

Step 3: Evaluate the function at the midpoints.

Substituting the midpoints into the function f(x) = x2, we get the following values:

f(0.5) = (0.5)2 = 0.25

f(1.5) = (1.5)2 = 2.25

f(2.5) = (2.5)2 = 6.25

f(3.5) = (3.5)2 = 12.25

 

Step 4: Calculate the areas of the rectangles.

Multiply the height of each rectangle (function value) by the width of each subinterval.

Area of rectangle 1 = 0.25 * 1 = 0.25

Area of rectangle 2 = 2.25 * 1 = 2.25

Area of rectangle 3 = 6.25 * 1 = 6.25

Area of rectangle 4 = 12.25 * 1 = 12.25

 

Step 5: Sum up the areas of the rectangles.

Total area = Area of rectangle 1 + Area of rectangle 2 + Area of rectangle 3 + Area of rectangle 4

Total area = 0.25 + 2.25 + 6.25 + 12.25 = 21

 

Therefore, the midpoint Riemann sum approximation of the integral of f(x) = x2 on the interval [0, 4] with four subintervals is 21.

 

Example 2:   Approximate the integral of the function g(x) = 2x + 1 on the interval [1, 5] using six subintervals and the right Riemann sum.

 

Solution:

Step 1: Determine the width of each subinterval.

The interval [1, 5] is divided into six equal subintervals, so the width of each subinterval is (5-1)/6 = 0.67 (rounded to two decimal places).

 

Step 2: Calculate the right endpoints of the subintervals.

The right endpoints of the subintervals are 1.67, 2.33, 3.00, 3.67, 4.33, and 5.00.

 

Step 3: Evaluate the function at the right endpoints.

Substituting the right endpoints into the function g(x) = 2x + 1, we get the following values:

g(1.67) = 2(1.67) + 1 = 4.34

g(2.33) = 2(2.33) + 1 = 5.66

g(3.00) = 2(3.00) + 1 = 7.00

g(3.67) = 2(3.67) + 1 = 8.34

g(4.33) = 2(4.33) + 1 = 9.66

g(5.00) = 2(5.00) + 1 = 11.00

 

Step 4: Calculate the areas of the rectangles.

Multiply the height of each rectangle (function value) by the width of each subinterval.

Area of rectangle 1 = 4.34 * 0.67 = 2.90 (rounded to two decimal places)

Area of rectangle 2 = 5.66 * 0.67 = 3.79 (rounded to two decimal places)

Area of rectangle 3 = 7.00 * 0.67 = 4.69 (rounded to two decimal places)

Area of rectangle 4 = 8.34 * 0.67 = 5.59 (rounded to two decimal places)

Area of rectangle 5 = 9.66 * 0.67 = 6.49 (rounded to two decimal places)

Area of rectangle 6 = 11.00 * 0.67 = 7.37 (rounded to two decimal places)

 

Step 5: Sum up the areas of the rectangles.

Total area = Area of rectangle 1 + Area of rectangle 2 + Area of rectangle 3 + Area of rectangle 4 + Area of rectangle 5 + Area of rectangle 6

Total area = 2.90 + 3.79 + 4.69 + 5.59 + 6.49 + 7.37 = 30.83 (rounded to two decimal places)

 

Therefore, the right Riemann sum approximation of the integral of g(x) = 2x + 1 on the interval [1, 5] with six subintervals is approximately 30.83.

Key Points

  • Riemann sums are methods used to approximate the area under a curve using rectangles.
  • The interval of integration is divided into subintervals of equal width.
  • The width of each subinterval is determined by partitioning the interval.
  • Riemann sums can use different endpoint rules: left, right, or midpoint, to determine the height of the rectangles.
  • Left Riemann sums use the left endpoint of each subinterval to determine rectangle heights.
  • Right, Riemann sums use the right endpoint of each subinterval to determine rectangle heights.
  • Midpoint Riemann sums use the midpoint of each subinterval to determine rectangle heights.
  • The height of each rectangle is determined by evaluating the function at the chosen endpoint or midpoint.
  • The area of each rectangle is calculated by multiplying the height by the width of the subinterval.
  • The areas of all the rectangles are then summed up to approximate the total area under the curve.
  • Increasing the number of subintervals improves the accuracy of the approximation.
  • The limit of Riemann sums as the number of subintervals approaches infinity yields the exact value of the definite integral.

Most Read

Unit: Application of Integrations Chapter: Area between Curves & Different Methods Reference: – Behaviour of a function, Different types of Asymptotes, Continuity of a function, Piecewise functions, Existence of Root Methods, Infinite & Asymptote limits, Bisection methods, Oblique Functions, Vertical Asymptotes, Exponential functions, Logarithmic limits, Squeeze theorem, Intermediate value theorem, Rational Functions After studying this […]

Unit: Application of Integrations Chapter: Modeling Particle Motion & Accumulation Problems Reference: – Motion Particle, Differentiability & Continuity, Increasing & Decreasing functions, Curve sketching, Analysis of a function, Optimization problems, First & Second Derivative test, Related rates, Local extrema, Implicit differentiation & Applications. After studying this chapter, you should be able to: Purpose of topic […]

Unit: Application of Integrations Chapter: Value of Function by Definite Integral Reference: – Differentiability & Continuity, Increasing & Decreasing functions, Curve sketching, Analysis of a function, Optimization problems, First & Second Derivative test, Related rates, Local extrema, Implicit differentiation & Applications.   After studying this chapter, you should be able to: Identifying the Function & […]