Trapezoidal Rule Calculator

what's the Trapezoidal Rule?

In numerical analysis, the trapezoidal rule is a technique for estimating the specific fundamental.

∫^x_y f(y) dy

The trapezoid rule works with the aid of estimating the area beneath the graph of a feature f(y) as a trapezium and computing its place with:

∫^x_y f(j) dj = ( x – y) . f(x) + f(y) / 2

The trapezoidal rule calculator used the Trapezium method to estimate the exact integrals.

A way to remedy quintessential characteristic using Trapezoidal Rule?

Observe these tips to calculate any feature region the use of trapezoidal rule manually.

• First, select a characteristic with integration and decrease, upper limits.
• Now, trapezoidal sum components to determine the trapezoid rule of the feature is

$$∫^x_y f(a)da ≈ Δa/2 [f(a^0) + 2f(a^1) +. . . +2f(a^{n-1}) + f(x^n), \text { where} , Δa = (y-x)/n.$$

• Divide the c programming language into smaller trapezoids of period Δa.
• Inside the subintervals final one is y and the first one is x, and examine the functions at the ones trapezoids values.
• Plug the values within the trapezoidal rule method and sum up the obtained values to get the predicted fee.

Example:

Use the Trapezoidal Rule with \( n = 4 \) to estimate:

$$ \int_{x=0}^{y=2} \sqrt{1 + \cos(2x)} \, dx $$

Solution:

The function \( f(x) \) is:

$$ f(x) = \sqrt{1 + \cos(2x)} $$

Given: \( x = 0 \), \( y = 2 \), \( n = 4 \).

According to the Trapezoidal Rule:

$$ \int_x^y f(x) \, dx \approx \frac{\Delta x}{2} \left[ f(x^0) + 2f(x^1) + \dots + 2f(x^{n-1}) + f(x^n) \right] $$

Calculate \( \Delta x \):

$$ \Delta x = \frac{y - x}{n} = \frac{2 - 0}{4} = 0.5 $$

Divide the interval \([0, 2]\) into \( n = 4 \) subintervals of length \( \Delta x = 0.5 \), with endpoints:

$$ x = 0, 0.5, 1.0, 1.5, 2.0 $$

Evaluate the function at these endpoints:

$$ f(x^0) = f(0) = \sqrt{1 + \cos(2(0))} = \sqrt{1 + 1} = \sqrt{2} \approx 1.4142 $$

$$ 2f(x^1) = 2f(0.5) = 2 \cdot \sqrt{1 + \cos(2(0.5))} = 2 \cdot \sqrt{1 + \cos(1)} \approx 2 \cdot 1.4978 = 2.9956 $$

$$ 2f(x^2) = 2f(1.0) = 2 \cdot \sqrt{1 + \cos(2(1.0))} = 2 \cdot \sqrt{1 + \cos(2)} \approx 2 \cdot 1.3254 = 2.6508 $$

$$ 2f(x^3) = 2f(1.5) = 2 \cdot \sqrt{1 + \cos(2(1.5))} = 2 \cdot \sqrt{1 + \cos(3)} \approx 2 \cdot 1.0819 = 2.1638 $$

$$ f(x^4) = f(2.0) = \sqrt{1 + \cos(2(2.0))} = \sqrt{1 + \cos(4)} \approx 1.4162 $$

Now calculate the sum:

$$ \Delta x / 2 = 0.5 / 2 = 0.25 $$

$$ \int_x^y f(x) \, dx \approx 0.25 \cdot (1.4142 + 2.9956 + 2.6508 + 2.1638 + 1.4162) $$

$$ \approx 0.25 \cdot 10.6406 = 2.6602 $$

Final Answer: The approximate value of the integral \( \int_{0}^{2} \sqrt{1 + \cos(2x)} \, dx \) using the Trapezoidal Rule is \( 2.6602 \).

Here you can also verify the results using an online trapezoidal rule calculator for quick computation.

FAQ:

Is the trapezium rule the same as Simpson's rule?

Simpson's Rule may be called Parabolic Rule. but, the trapezoidal rule may be referred to as the Trapezoid rule.

Which rule is higher trapezoidal or Simpsons?

while we work with quadratic features, the Simpsons rule offers the exceptional estimation and the Trapezoidal supplied the worst approximation. As properly, for the trigonometric features, the Simpsons approach also gave the most correct estimation even as the Trapezoid gave the least accurate estimation.