help me to answer this question (integrals)?

Question

there is a question asking to estimate the integral from 0 to 1 of 1/(x^2+1) dx using the left hand sum, right hand sum ,and the trapezoidal rule. I have obtained those answer's.

and there is the part b question where it asks if the exact value of the integral is phi/4, then estimate the value of phi using part(a). I dont really understand this question does it mean that if in part a I got an answer of 0.86 using the left hand sum then phi is 0.86/4??? helpp meee

Answer 1

Well, no, if you got 0.86 it would be 0.86 * 4. Think about it, x = integral = phi/4, thus x = phi/4 and 4x = phi.

Answer 2

420

Answer 3

I'm not sure what you need since I don't have the full question in front of me. But here is the evaluation of the integral on [0,1].

∫dx/(x² + 1)

Let
tan θ = x
sec²θ dθ = dx
θ = arctan x

∫dx/(x² + 1) = ∫(sec² θ dθ)/(tan²θ + 1) = ∫(sec² θ dθ)/sec²θ
= ∫dθ = θ = arctan x | on [0,1] = (arctan 1) - (arctan 0) = π/4 - 0 = π/4