Calculating the Function Values

Oct 13, 2025 by magicalceylontours.com 105 views

Introduction to Simpson's Rule

Simpson's Rule is a numerical method used to approximate the value of a definite integral. It is a powerful tool for solving problems in mathematics, physics, and engineering. In this article, we will explore how to use Simpson's Rule with $n = 4$ steps to estimate the integral $\int_0^2 3x^2 \, dx$ .

Understanding Simpson's Rule

Simpson's Rule is based on the idea of approximating the area under a curve by dividing it into small parabolic segments. The rule states that the integral of a function $f(x)$ from $a$ to $b$ can be approximated as:

\int_a^b f(x) \, dx \approx \frac{h}{3} \left[ f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + \cdots + 2f(x_{n-2}) + 4f(x_{n-1}) + f(x_n) \right]

where $h = \frac{b-a}{n}$ is the width of each subinterval, and $x_i = a + ih$ are the points at which the function is evaluated.

Applying Simpson's Rule to the Given Integral

To estimate the integral $\int_0^2 3x^2 \, dx$ using Simpson's Rule with $n = 4$ steps, we need to divide the interval $[0, 2]$ into four equal subintervals. The width of each subinterval is $h = \frac{2-0}{4} = \frac{1}{2}$ .

We then evaluate the function $f(x) = 3x^2$ at the points $x_0 = 0$ , $x_1 = \frac{1}{2}$ , $x_2 = 1$ , $x_3 = \frac{3}{2}$ , and $x_4 = 2$ .

Calculating the Function Values

We calculate the function values as follows:

$f(x_0) = f(0) = 3(0)^2 = 0$
$f(x_1) = f\left(\frac{1}{2}\right) = 3\left(\frac{1}{2}\right)^2 = \frac{3}{4}$
$f(x_2) = f(1) = 3(1)^2 = 3$
$f(x_3) = f\left(\frac{3}{2}\right) = 3\left(\frac{3}{2}\right)^2 = \frac{27}{4}$
$f(x_4) = f(2) = 3(2)^2 = 12$

Applying Simpson's Rule

We now apply Simpson's Rule to estimate the integral:

\int_0^2 3x^2 \, dx \approx \frac{h}{3} \left[ f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + f(x_4) \right]

Substituting the values, we get:

\int_0^2 3x^2 \, dx \approx \frac{\frac{1}{2}}{3} \left[ 0 + 4\left(\frac{3}{4}\right) + 2(3) + 4\left(\frac{27}{4}\right) + 12 \right]

Simplifying, we get:

\int_0^2 3x^2 \, dx \approx \frac{1}{6} \left[ 3 + 6 + 27 + 12 \right]

\int_0^2 3x^2 \, dx \approx \frac{1}{6} \left[ 48 \right]

\int_0^2 3x^2 \, dx \approx 8

Conclusion

In this article, we used Simpson's Rule with $n = 4$ steps to estimate the integral $\int_0^2 3x^2 \, dx$ . We divided the interval $[0, 2]$ into four equal subintervals and evaluated the function at the points $x_0 = 0$ , $x_1 = \frac{1}{2}$ , $x_2 = 1$ , $x_3 = \frac{3}{2}$ , and $x_4 = 2$ . We then applied Simpson's Rule to estimate the integral, and obtained an approximate value of $8$ .

The final answer is $\boxed{8}$ .

Introduction

Simpson's Rule is a powerful numerical method used to approximate the value of a definite integral. In our previous article, we explored how to use Simpson's Rule with $n = 4$ steps to estimate the integral $\int_0^2 3x^2 \, dx$ . In this article, we will answer some frequently asked questions about Simpson's Rule.

Q: What is Simpson's Rule?

A: Simpson's Rule is a numerical method used to approximate the value of a definite integral. It is based on the idea of approximating the area under a curve by dividing it into small parabolic segments.

Q: How does Simpson's Rule work?

A: Simpson's Rule works by dividing the interval $[a, b]$ into $n$ equal subintervals, and evaluating the function at the points $x_0 = a$ , $x_1 = a + h$ , $x_2 = a + 2h$ , ..., $x_n = a + nh$ , where $h = \frac{b-a}{n}$ . The rule then approximates the integral as:

\int_a^b f(x) \, dx \approx \frac{h}{3} \left[ f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + \cdots + 2f(x_{n-2}) + 4f(x_{n-1}) + f(x_n) \right]

Q: What are the advantages of Simpson's Rule?

A: Simpson's Rule has several advantages, including:

It is more accurate than the Trapezoidal Rule for a given number of subintervals.
It is more efficient than the Trapezoidal Rule for a given level of accuracy.
It can be used to approximate the value of a definite integral with a high degree of accuracy.

Q: What are the disadvantages of Simpson's Rule?

A: Simpson's Rule has several disadvantages, including:

It requires more function evaluations than the Trapezoidal Rule.
It is more complex to implement than the Trapezoidal Rule.
It may not be as accurate as other numerical methods, such as Gaussian quadrature.

Q: When should I use Simpson's Rule?

A: You should use Simpson's Rule when:

You need to approximate the value of a definite integral with a high degree of accuracy.
You have a function that is smooth and well-behaved.
You have a large number of subintervals.

Q: How do I choose the number of subintervals?

A: You should choose the number of subintervals based on the desired level of accuracy. A larger number of subintervals will generally result in a more accurate approximation.

Q: Can I use Simpson's Rule with an odd number of subintervals?

A: Yes, you can use Simpson's Rule with an odd number of subintervals. However, you will need to use a modified version of the rule, known as Simpson's 3/8 Rule.

Q: Can I use Simpson's Rule with a non-uniform grid?

A: No, Simpson's Rule is designed to work with a uniform grid. If you need to use a non-uniform grid, you may need to use a different numerical method.

Conclusion

In this article, we answered some frequently asked questions about Simpson's Rule. We discussed the advantages and disadvantages of Simpson's Rule, and provided guidance on when to use it and how to choose the number of subintervals. We also discussed some common variations of Simpson's Rule, including Simpson's 3/8 Rule and Gaussian quadrature.

The final answer is $\boxed{8}$ .