Use series to approximate the definite integral to within the indicated accuracy.?

Question

integral (0,.5) cos x^2 dx (SHOW WORK) Give your answer correct to 3 decimal places.

Answer 1

First let us note that cos u expands in a Taylor/McLauren series to:

cos u = 1 - (1/2!)u^2 + (1/4!)u^4 ...

So I[0.0, 0.5] of cos x^2 dx corresponds to:
(I[0.0, 0.5] of 1) - (I[0.0, 0.5] of (1/2!)u^2) + ...

Computing the integral of a monomial is straightforward.

We can substitute u = x^2 and still get a polynomial in x, so the basic idea of splitting the integral into a series still works.

Next we note that if x <= 0.5,
u < 0.25 and u^n <= (0.25)^n

This allows us to get an upper bound on the size of each term.

Since we have an alternating series, the error is no worse than the first term dropped.

So we compute the definite integrals, one at a time until our estimate of the value of the next shows that it is not relevant.