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Wize University Calculus 1 Textbook > Integration Techniques

Simpson's Rule

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Simpson's Rule

We can begin to get more creative with our Reimann Sums to get better approximations for area under curves. Instead of using a rectangle approximation, we can place quadratics on the graph to approximate area.

Simpson's Rule Approximation

∫abf(x)dx ≈Δx3(f(x0)+4f(x1)+2f(x2)+4f(x3)+2f(x4)+...+4f(xn−1)+f(xn))\boxed{\displaystyle \int_a^bf\left(x\right)dx\ \approx\frac{\Delta x}{3}\left(f\left(x_0\right)+4f\left(x_1\right)+2f\left(x_2\right)+4f\left(x_3\right)+2f\left(x_4\right)+...+4f\left(x_{n-1}\right)+f\left(x_n\right)\right)}

where Δx=b−an,  xi=a+iΔx\displaystyle\Delta x=\frac{b-a}{n}, \ \ x_i=a+i\Delta x, and nn is the number of subintervals.

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Example: Simpson's Rule

Use a Simpson's Rule approximation for

∫04ex2dx\displaystyle \int_0^4e^{x^2}dx

using 4 sub intervals.


∫abf(x)dx ≈Δx3(f(x0)+4f(x1)+2f(x2)+4f(x3)+2f(x4)+...+4f(xn−1)+f(xn))\displaystyle \int_a^bf\left(x\right)dx\ \approx\frac{\Delta x}{3}\left(f\left(x_0\right)+4f\left(x_1\right)+2f\left(x_2\right)+4f\left(x_3\right)+2f\left(x_4\right)+...+4f\left(x_{n-1}\right)+f\left(x_n\right)\right)


Δx=b−an=4−04=1\displaystyle \Delta x=\frac{b-a}{n}=\frac{4-0}{4}=1

∫04ex2dx≈13(f(0)+4f(1)+2f(2)+4f(3)+f(4)) =13(e02+4e12+2e22+4e32+e42)\displaystyle \int_0^4e^{x^2}dx\approx\frac{1}{3}\left(f\left(0\right)+4f\left(1\right)+2f\left(2\right)+4f\left(3\right)+f\left(4\right)\right)\ =\frac{1}{3}\left(e^{0^2}+4e^{1^2}+2e^{2^2}+4e^{3^2}+e^{4^2}\right)

=13(1+4e+2e4+4e9+e16)=\frac{1}{3}\left(1+4e+2e^4+4e^9+e^{16}\right)
Using the Simpson's rule with n=4n=4, find the integral approximation of

∫04x2dx\displaystyle \int_0^4 x^2 dx.