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Effectus [21]
3 years ago
13

Apply Gaussian quadrature with n = 4 to approximate integrate sin x^2dx from 1 to 5

Mathematics
1 answer:
maks197457 [2]3 years ago
4 0
First, recall that Gaussian quadrature is based around integrating a function over the interval [-1,1], so transform the function argument accordingly to change the integral over [1,5] to an equivalent one over [-1,1].

x=2t+3\iff t=\dfrac x2-\dfrac32\implies2\mathrm dt=\mathrm dx
x=1\implies t=\dfrac{2-6}4=-1
x=5\implies t=\dfrac{10-6}4=1

So,

\displaystyle\int_{x=1}^{x=5}\sin x^2\,\mathrm dx=\displaystyle2\int_{t=-1}^{t=1}\sin(2t+3)^2\,\mathrm dt

Let f(t)=2\sin(2t+3)^2. With n=4, we're looking for coefficients c_i and nodes x_i, with 1\le i\le4, such that

\displaystyle\int_{-1}^1f(t)\,\mathrm dt\approx c_1f(x_1)+\cdots+c_4f(x_4)

You can either try solving for each with the help of a calculator, or look up the values of the weights and nodes (they're extensively tabulated, and I'll include a link to one such reference).

Using the quadrature, we then have

\displaystyle\int_{-1}^1f(t)\,\mathrm dt\approx0.3749f(-0.8611)+0.6521f(-0.3400)+0.6521f(0.3400)+0.3749f-0.8611)
\displaystyle\int_{-1}^1f(t)\,\mathrm dt\approx0.5790
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