Question

Use a Taylor series to approximate the following definite integrals. Retain as many terms as needed to ensure the error is less than 10−4. ∫0.201+x4−−−−−√dx

Answer 1

I= 0.40102036

Explanation:

∧ = raise to power

Given the equation

F(x) = √x² + 1.0

Now recalling the Taylor Series in respect to the approximation for degree n at x = a....

Therefore, it is given by :

f(x) = f(a) + f° (a) ((x - a)) + ( 1/2) f°° (a)((x -a)² + .....(1/n!) f∧n (a) ((x - a) ∧ n

Referring back to the Taylor Series

f(x) = - x ∧8/8 + x ∧ 4/2 + 1

If we integrate the Taylor series, we get,

f°(x) = - x ³*³/72 + x∧5/10 + x

At this point, we apply limit (lim) to the integrals

I = ∫ lim(0.4) (0) f(x) dx

I = ∫ (- x ∧8/8 + x ∧ 4/2 +1)dx

I = [ ( -x³*³/72 + x∧5/10 +x)]lim(0.4) lim(0)

The final answer then is:

I= 0.40102036