simpsons rule pdf
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h. Its strength is that, although rectangles and trapezoids work better for linear functions a formula for the area of a trapezoid. where f (x) is called the integrand, = lower limit of integration. That is, approximate the definite integral ∫x dx by Simpson’s Rule Secon order Newton-Cotes formula is called Simpson’s Rule. = upper limit of integration Notes on Simpson's Rule Background The idea of Simpson's Rule is to approximate a definite integral as’ follows: +, 0—BÑ.BSubdivide the interval [a,b] into n subintervals. We do the same for f(x) = x Simpson’s Rule combines this notion into a formula which weights the two according to their relative errors: ()M T f x dx b a + ∫ ≅. Here’s a good way to derive Simpson’s formula. The derivation of the formula for Simpson’s Rule is left as an exercise, but the area of this region is essentially the base times some average height of thegraph: yo + 4y1 + yArea = (base)(average height) = (2Δx)This emphasizes the middle more than the sides, which is To guarantee this with the Trapezoidal Rule, we could take n big enough so that 1=n, or n So n = would work. Make sure n is evenCorresponding to the usual,,,BBB!"#8ÆB, notation for the endpoints of the subintervals of [+, ], (i.e.,B a. Example A, Simpson’s Rule Notes on Simpson's Rule Background The idea of Simpson's Rule is to approximate a definite integral as’ follows: +, 0—BÑ.BSubdivide the interval [a,b] into n subintervals Derivation of Simpson’s Rule by attempting to cancel errors Consider f(x)=xOn the one hand Rxdx = x=On the other hand the Trapezoid approximation is Tn To guarantee this with the Trapezoidal Rule, we could take n big enough so that 1=n, or n So n = would work. Formally, we should consider the node points xi, xi+1 = xi + h, and xi+2 = xi + 2h and the corresponding values of the function fi, fi+1, and fi+But to simplify the algebra, let us instead consider x = 0, 1,and label the FigureUsing a parabolic approximation of the curve. = (b−a)+2+which is (b−a), so the formula works for f(x) =b. xxxxxxxxxx Note the very steep slope of the function y =ex2 when x >At x = Extended Simpson’s 1/3 Rule Simply add up integrated values obtained using Simpson’s 1/3 rule over each sub-interval• Sub-interval size = Number of sub Simpson’s Rule Simpson’s Rule, named after Thomas Simpson though also used by Kepler a century before, was a way to approximate integrals without having to deal with lots of narrow rectangles (which also implies lots of imal calculations). Formally, we should consider the node points xi, xi+1 = xi + h, and Simpson’s Rule combines this notion into a formula which weights the two according to their relative errors: ()M T f x dx b a + ∫ ≅. But to guarantee this with Simpson’s Rule, it would Department of Mathematics University of Washington Evaluate the integrals using Simpson’s Rule. With reference to the sketch above, the area of a trapezoid consists of the area of the rectangle plus the area of the triangle, or hyL + (h=2)(yR yL) = h(yL + yR)=So, the area is h times the average of the lengths of the two vertical edges Here, we will discuss Simpson’s 1/3 rule of integral approximation, which improves upon the accuracy of the trapezoidal rule. yL. Here, we will discuss the Simpson’s 1/3 rule of approximating integrals of the form. In this case, a=0, b=5, and use the ten steps from 0 Example C: Approximate ∫ex dx using Simpson’s Rule for n =∆x =? Example A, Simpson’s Rule: Approximate the area under the curve y = x on the interval≤ x ≤usingsubintervals. yRyL yR. But to guarantee this with Simpson’s Rule, it would su ce to choose n so thatn4, or nFor this, n =almost su ces, and we certainly could get the desired accuracy with n = 8 The accuracy of the Simpson’s Rule approximation of an integral increases as you use more strips. If f(x) =then Z b a f(x)dx = (b−a) and hf(a)+2f(m)+ f(b) 2! Here’s a good way to derive Simpson’s formula. ExampleUsing Simpson’s rule, find the value of Ze2x sin(x) dx Secon order Newton-Cotes formula is called Simpson’s Rule. = ∫ f (x) dx.
