Trapezoidal Rule Calculator simply requires input function, range and number of trapezoids in the specified input fields to get the exact results within no time. So, enter your details in the below input box and click on the calculate button to get the answer in fraction of seconds.

**Trapezoidal Rule Calculator: **No need to feel solving any function using trapezoidal rule is a bit difficult. Our handy calculator tool gives the answer easily and quickly. Along with this free calculator, you can also get the detailed explanation to solve the integration functions using trapezoidal rule. Have a look at the example, trapezoidal rule definition and formula in the below sections.

Follow these simple and easy guidelines to solve any function integration using trapezoidal rule manually.

- Take any function f(x) with integration and lower, upper limits i.e a,b.
- Also, know the number of trapezoids n.
- The formula to compute trapezoidal rule of any function is
- ∫
_{a}^{b}f(x)dx ≈ Δx/2 [f(x0) + 2f(x1) + 2f(x2)+. . . +2f(xn-1) + f(xn) - Where, Δx = (b-a)/n.
- Divide the given interval into n portions of length Δx.
- In the subintervals last one is b and first one is a.
- Evaluate the functions at those subinterval values.
- Substitute the obtained values in the formula of trapezoidal rule and sum up the values to get the approximate value.

**Example**

**Question: Use the Trapezoidal Rule with n = 5 to approximate ∫ _{a=0}^{b=1} √(1+sin^{3}(x).**

**Solution:**

Given that,

function f(x) = √(1+sin^{3}(x)

a= 0, b= 1, n=5.

Trapezoidal Rule dtates that

∫_{a}^{b} f(x)dx ≈ Δx/2 [f(x0) + 2f(x1) + 2f(x2)+. . . +2f(xn-1) + f(xn), where

Δx = (b-a)/n

Substitute the given values in above formula to get Δx value.

Δx = 1-0/5 = 1/5

Divide the interval [0,1] into n=5 subintervals of length Δx=1/5, with the following endpoints:

a = 0, 1/5, 2/5, 3/5, 4/5, 1= b

Evaluate the function at these end points:

f(x0) = f(a) = f(0) = √(1+sin^{3}(0) = 1

2f(x1) = 2f(1/5) = 2√(1+sin^{3}(1/5) = 2.00782606791279

2f(x2) = 2f(2/5) = 2√(1+sin^{3}(2/5) = 2.05820697233265

2f(x3) = 2f(3/5) = 2√(1+sin^{3}(3/5) = 2.17257446116512

2f(x4) = 2f(4/5) = 2√(1+sin^{3}(4/5) = 2.34021475342487

f(x5) = f(1) = 2√(1+sin^{3}(1) = 1.26325897447473

Δx/2 = 1/10

Finally sum up the above values and multiply by Δx/2

= ⅒ (1 + 2.00782606791279 + 2.05820697233265 + 2.17257446116512 + 2.34021475342487 + 1.26325897447473)

= 1.08420812293102

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**1. What is meant by Trapezoid Rule?**

Trapezoid Rule is to find the exact value of a definite integral using a numerical method. This rule is based on the Newton-Cotes formula which states that one can get the exact value of the integral as an nth order polynomial. Trapezium rule works by approximating the region under the graph of function as a trapezoid and calculating its area.

**2. Where do we use Trapezoid Rule?**

Trapezoid Rule is a rule that is used to determine the area under the curve. In this method, the area under the curve by dividing the total area into smaller trapezoids instead of dividing into rectangles. Integration method works by approximating the area under the graph of a function as a trapezoid and it calculates the area.

**3. What is the Trapezoid Rule Formula?**

Trapezoidal Rule formula is mentioned here:

∫_{a}^{b} f(x)dx ≈ Δx/2 [f(x0) + 2f(x1) + 2f(x2)+. . . +2f(xn-1) + f(xn),

where

Δx = (b-a)/n

**4. Approximate the integral ∫ _{0}^{1} x^{3} dx using the Trapezoidal Rule with n = 2 subintervals.**

Trapezoidal Rule formula with n = 2 subintervals is

T = Δx/2 [f(x0) + 2f(x1) + f(x2)]

Δx = (b-a)/n

=1-0/2 = 1/2

f(x0) = f(0) = 0

f(x1) = f(1/2) = (1/2)^{3} = 1/8

f(x2) = f(1) = 1^{3} = 1

So,

∫_{0}^{1} x^{3} dx ≈ 1/4 [0+ 2x(1/8) + 1] = 1/4 x 5/4 = 5/16