Trapezoidal Rule Formula

What is Trapezoidal rule?

The Trapezoidal Rule is a numerical method for approximating the definite integral of a function. It is based on approximating the area under a curve by dividing it into a series of trapezoids and summing up their areas.

Basic idea behind Trapezoidal rule:

The basic idea behind the Trapezoidal Rule is to approximate the function by a series of straight line segments connecting consecutive points on the curve. By considering each pair of adjacent points, we can construct a trapezoid with a base formed by the line segment connecting the two points and a height equal to the difference in their function values. The area of each trapezoid is then calculated using the formula for the area of a trapezoid:

Area = (base1 + base2) * height / 2

How to apply Trapezoidal rule?

To apply the Trapezoidal Rule, we divide the interval of integration [a, b] into smaller subintervals of equal width, where the width is denoted by h. The more subintervals we use, the closer our approximation will be to the exact value of the integral. The Trapezoidal Rule formula can be expressed as:

[Equation]f(x) dx ≈ h/2 * [f(a) + 2f(x₁) + 2f(x₂) + … + 2f(xₙ₋₁) + f(b)]

In this formula,

• f(a) and f(b) represent the function values at the endpoints of the interval
• f(x₁), f(x₂), …, f(xₙ₋₁) represent the function values at the intermediate points within the subintervals.

The sum of the function values inside the brackets, excluding the endpoints, is multiplied by 2 because these points contribute to the areas of two adjacent trapezoids. The factor h/2 is used to scale the sum appropriately.

By summing up the areas of all the trapezoids, the Trapezoidal Rule provides an approximation of the definite integral of the function over the interval [a, b].

Solved Examples on Trapezoidal Rule Formula

Example 1: Approximate the value of the definite integral ∫[0, 4] (2x + 1) dx using the Trapezoidal Rule with 4 subintervals.

Solution:

In this case, we have the function f(x) = 2x + 1 and we want to integrate it over the interval [0, 4] using 4 subintervals.

First, we need to calculate the width of each subinterval:

h = (b – a) / n

= (4 – 0) / 4

= 1

Now, we can apply the Trapezoidal Rule formula:

[Equation] (2x + 1) dx ≈ h/2 * [f(a) + 2f(x₁) + 2f(x₂) + 2f(x₃) + f(b)]

≈ 1/2 * [f(0) + 2f(1) + 2f(2) + 2f(3) + f(4)]

Calculating the function values at each point:

f(0) = 2(0) + 1 = 1

f(1) = 2(1) + 1 = 3

f(2) = 2(2) + 1 = 5

f(3) = 2(3) + 1 = 7

f(4) = 2(4) + 1 = 9

Plugging in the values:

[Equation] (2x + 1) dx ≈ 1/2 * [1 + 2(3) + 2(5) + 2(7) + 9]

≈ 1/2 * [1 + 6 + 10 + 14 + 9]

≈ 1/2 * 40 = 20

Therefore, the approximate value of the integral using the Trapezoidal Rule with 4 subintervals is 20.

Example 2: Approximate the value of the definite integral [Equation] (1/x) dx using the Trapezoidal Rule with 6 subintervals.

Solution:

In this case, we have the function f(x) = 1/x and we want to integrate it over the interval [1, 5] using 6 subintervals.

First, let’s calculate the width of each subinterval:

h = (b – a) / n

= (5 – 1) / 6

= 2/3

Now, we can apply the Trapezoidal Rule formula:

[Equation] (1/x) dx ≈ h/2 * [f(a) + 2f(x₁) + 2f(x₂) + … + 2f(x₅) + f(b)]

≈ (2/3)/2 * [f(1) + 2f(x₁) + 2f(x₂) + … + 2f(x₅) + f(5)]

Calculating the function values at each point:

f(1) = 1/1 = 1

f(x₁) = 1/x₁, where

x₁ = 1 + (2/3) = 5/3 ≈ 1.67

f(x₂) = 1/x₂, where

x₂ = 1 + 2(2/3) = 7/3 ≈ 2.33

Frequently Asked Questions on Trapezoidal Rule Formula

1: How do you determine the number of subintervals to use with the Trapezoidal Rule?

Answer: The number of subintervals is determined by the desired level of accuracy. As a general guideline, using more subintervals will result in a more accurate approximation. However, there is a trade-off between accuracy and computational cost, so the number of subintervals should be chosen based on the specific problem and available computational resources.

2: Can the Trapezoidal Rule handle functions with sharp curves or irregular behavior?

Answer: The Trapezoidal Rule can provide reasonable approximations for functions with moderate nonlinear behavior. However, for functions with sharp curves, rapid changes, or irregular behavior, the Trapezoidal Rule may not be accurate enough. In such cases, other numerical integration methods like Simpson’s Rule or Gaussian quadrature may be more appropriate.

3: What is 5 point trapezoidal rule?

Answer: The 5-point Trapezoidal Rule, also known as the Composite Trapezoidal Rule with 5 points, is an extension of the basic Trapezoidal Rule that uses additional function evaluations to improve the accuracy of the approximation.

In the 5-point Trapezoidal Rule, the interval of integration is divided into smaller subintervals, and within each subinterval, the function is evaluated at five points to construct a more accurate trapezoidal approximation.

Let’s denote the interval of integration as [a, b]. The width of each subinterval is h = (b – a) / n, where n is the number of subintervals. The 5-point Trapezoidal Rule formula can be expressed as:

∫[a, b] f(x) dx ≈ (h/12) * [f(a) + 5f(x₁) + f(x₂) + 6f(x₃) + f(x₄) + 5f(x₅) + f(b)]

In this formula, f(a) and f(b) represent the function values at the endpoints of the interval, while f(x₁), f(x₂), f(x₃), f(x₄), and f(x₅) represent the function values at the intermediate points within each subinterval.

The factor h/12 is used to scale the sum of the function values appropriately, taking into account the wider base of the trapezoids formed by the additional points.

By evaluating the function at five points within each subinterval and summing up the areas of the trapezoids, the 5-point Trapezoidal Rule provides a more accurate approximation of the definite integral compared to the basic Trapezoidal Rule.

Note that as the number of subintervals increases, the accuracy of the approximation improves. The 5-point Trapezoidal Rule is a commonly used method to achieve better accuracy while maintaining a reasonable computational cost.

4: Which curve is used in trapezoidal rule?

Answer: The Trapezoidal Rule can be applied to approximate the definite integral of any continuous function. It is not specific to a particular type of curve or function.

The Trapezoidal Rule is a numerical integration method that works by approximating the area under a curve by dividing it into a series of trapezoids. It assumes that the function being integrated is piecewise linear between consecutive points and constructs trapezoids by connecting these points with straight lines.

Therefore, whether the curve is linear, quadratic, exponential, trigonometric, or any other continuous function, the Trapezoidal Rule can be used to approximate its definite integral.

5: What is the alternate name of the trapezoidal formula?

Answer: The alternate name of the trapezoidal formula is the Trapezium Rule. In some regions, particularly in British English, the term “trapezium” is used instead of “trapezoid” to refer to a quadrilateral with no parallel sides. As a result, the method is sometimes called the Trapezium Rule to reflect this difference in terminology. The Trapezium Rule and the Trapezoidal Rule refer to the same numerical integration method; they are just named differently in different regions.

6: How to derive trapezoidal rule?

Answer:The Trapezoidal Rule is a numerical integration technique used to approximate the definite integral of a function. The derivation of the Trapezoidal Rule involves approximating the area under a curve by dividing it into trapezoids.

Let’s consider a function f(x) that we want to integrate over the interval [a, b]. The Trapezoidal Rule estimates the integral by approximating the function with a series of trapezoids.

Step 1: Divide the interval [a, b] into n subintervals of equal width, denoted by h. The width of each subinterval is given by h = (b – a) / n.

Step 2: Approximate the function f(x) on each subinterval by connecting the endpoints with a straight line. This results in a series of trapezoids.

Step 3: Calculate the area of each trapezoid. The area of a trapezoid is given by the formula: Area = (base1 + base2) * height / 2, where base1 and base2 are the lengths of the two parallel bases of the trapezoid, and height is the perpendicular distance between the bases.

Step 4: Sum up the areas of all the trapezoids to obtain an approximation of the integral. The approximation is given by the formula:

Approximation = h * [(f(a) + f(b)) / 2 + ∑(f(xi))], where xi represents the x-values within each subinterval, excluding the endpoints.

The derivation of the Trapezoidal Rule involves approximating the function within each subinterval with a linear function and calculating the area of each trapezoid. The approximation becomes more accurate as the number of subintervals (n) increases, resulting in a smaller width (h) and a closer approximation to the actual integral.

7: What is the difference between Trapezoidal rule and Riemann Sums rule?

Answer: The Trapezoidal Rule divides the interval into subintervals and approximates the function within each subinterval with a straight line. It then calculates the area of each trapezoid formed by these approximations and sums them up to estimate the integral. The Trapezoidal Rule provides a better approximation to the integral than the simple Riemann Sum by considering the average of the function values at the endpoints of each subinterval.

On the other hand, Riemann Sums approximate the area under the curve by dividing the interval into subintervals and using the height of rectangles to estimate the function values. The most commonly used Riemann Sum is the left Riemann Sum, which takes the left endpoint of each subinterval as the height of the rectangle. Other variations include the right Riemann Sum and the midpoint Riemann Sum. Riemann Sums provide a basic approximation to the integral and become more accurate as the number of subintervals increases.

8: What is n in trapezoidal rule?

Answer: In the Trapezoidal Rule, ‘n’ represents the number of subintervals into which the interval [a, b] is divided. The larger the value of ‘n’, the more subintervals there are, resulting in a more accurate approximation of the definite integral. Increasing ‘n’ decreases the width of each subinterval, allowing for a better approximation of the curve within each interval. As ‘n’ approaches infinity, the Trapezoidal Rule converges to the exact value of the integral.