# What is Integration Using Simpson's Rule?

Simpson's Rule will give you a better approximation of the integral than the other basic methods.

The other methods are Rectangular Approximation Method (RAM) - left, middle, and right; and the Trapezoidal Rule.

Numerical integration is used when we are given a set of data (evenly spaced on the independent variable) rather than an explicit function. This happens when data is collected from some measuring device.

Numerical integration is also used when it is difficult to find the antiderivative of the integrand.

And of course numerical integration is used on integrable functions to learn how it works.

Simpson's Rule is more accurate than the other methods because they use linear structures (rectangles and trapezoids) to approximate. Simpson's Rule uses quadratics (parabolas) to approximate. Most real-life functions are curves rather than lines, so Simpson's Rule gives the better result, unless the function that you are approximating is actually linear.

Simpson's Rule requires that the data set have an odd number of elements which gives you an even number of intervals. The other methods do not have this restriction.

Explains the subject in plain English can be found here: https://tutor.hix.ai

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Integration using Simpson's rule is a numerical method for approximating the definite integral of a function over a specified interval. It is based on approximating the function by a second-degree polynomial (a quadratic) between each pair of adjacent points, and then integrating these quadratic approximations over the interval. Simpson's rule is more accurate than the trapezoidal rule and is particularly effective for smoothly varying functions.

The formula for Simpson's rule can be expressed as:

[ \int_{a}^{b} f(x) , dx \approx \frac{h}{3} [f(a) + 4f(a + h) + 2f(a + 2h) + 4f(a + 3h) + \ldots + 2f(b - h) + f(b)] ]

Where:

- (a) and (b) are the lower and upper limits of integration, respectively.
- (f(x)) is the function being integrated.
- (h) is the width of each subinterval, calculated as (\frac{b - a}{n}), where (n) is the number of subintervals.

Simpson's rule requires an even number of subintervals for its application. The more subintervals used, the more accurate the approximation tends to be.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- How do you Use Simpson's rule with #n=8# to approximate the integral #int_0^pix^2*sin(x)dx#?
- How do you approximate the area under #y=10−x^2# on the interval [1, 3] using 4 subintervals and midpoints?
- How do you write the Simpson’s rule and Trapezoid rule approximations to the #intsinx/x# over the inteval [0,1] with #n=6#?
- How do you use n = 5 equally divided subdivisions to estimate the integral from 1 to 3 of #(1/x^2)dx# with the right-hand Riemann sum?
- Determine a region whose area is equal to the given limit?

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