Verifying Int. of f(x) Using Trapezoid Rule

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In summary, the conversation is about verifying a solution and calculating the integral using the trapezoid rule. The data about the function f(x) is also provided, and the correct constant for the trapezoid rule is discussed. The final estimate of the integral is approximately 2.28025.
  • #1
shamieh
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Just need someone to verify my solution and that I have done my calculations correctly. Thank you again for all of your help in advance, everyone on this forum has helped me SO much with my classes this semester.

Using the trapezoid rule.

The following data was collected about the function \(\displaystyle f(x)\)

Estimate \(\displaystyle \int^{1.25}_0 f(x) \, dx\)

\(\displaystyle x | f(x)\)

\(\displaystyle 0 | 3.000\)

\(\displaystyle .25 | 2.540\)

\(\displaystyle .50 | 1.583\)

\(\displaystyle .75 | 1.010\)

\(\displaystyle 1.00 | 1.346\)

\(\displaystyle 1.25 | 2.284\)

I ended up with \(\displaystyle .25[3.000 + 2(2.540) + 2(1.583) + 2(1.010) + 2(1.346) + 2.284] \approx 4.5605\)
 
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  • #2
The constant in front should be:

\(\displaystyle \frac{b-a}{2n}=\frac{1.25-0}{2\cdot5}=0.125\)

Your result is twice as large as it should be. :D
 
  • #3
MarkFL said:
The constant in front should be:

\(\displaystyle \frac{b-a}{2n}=\frac{1.25-0}{2\cdot5}=0.125\)

Your result is twice as large as it should be. :D

I knew something didn't look right...lol
Thanks Mark!

- - - Updated - - -

so does $\approx 2.28025$ look more accurate?
 
  • #4
Yes, that looks good. :D
 

FAQ: Verifying Int. of f(x) Using Trapezoid Rule

What is the Trapezoid Rule used for?

The Trapezoid Rule is a numerical integration method used to approximate the area under a curve by dividing the interval into trapezoids and summing the areas of these trapezoids.

How do you use the Trapezoid Rule to verify the integral of a function?

To use the Trapezoid Rule to verify the integral of a function, you first need to divide the interval into smaller subintervals. Then, you use the formula for the area of a trapezoid to calculate the area under the curve for each subinterval. Finally, you sum up all the areas to get an approximation of the integral.

What are the advantages of using the Trapezoid Rule over other numerical integration methods?

One advantage of the Trapezoid Rule is that it is relatively simple to use and understand. It also tends to give more accurate results for functions that are not smooth or have sharp turns. Additionally, it can be easily implemented in a computer program.

Can the Trapezoid Rule be used to find exact values of integrals?

No, the Trapezoid Rule is only an approximation method and can never give exact values for integrals. However, by increasing the number of subintervals, you can get a more accurate approximation.

Are there any limitations of using the Trapezoid Rule?

Yes, the Trapezoid Rule is not suitable for all types of functions. It can give inaccurate results for functions with rapid changes or oscillations. Additionally, the accuracy of the approximation depends on the number of subintervals used, so it may be time-consuming to get a highly accurate result.

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