Question

this question has several parts that must be completed sequentially. if you skip a part of the question, you will not receive any points for the skipped part, and you will not be able to come back to the skipped part.
estimate
∫₀¹ 2 cos(x²) dx
using the following rule with n = 4.
exercise (a)
the trapezoidal rule
click here to begin!
exercise (b)
the midpoint rule
step 1
the midpoint rule says that
∫ₐᵇ f(x) dx = mₙ = δxf(ˉx₁) + f(ˉx₂) + … + f(ˉxₙ)
with δx = (b - a)/n.
we again need to estimate
∫₀¹ 2 cos(x²) dx
with n = 4 subintervals.
for this,
δx = (1 - 0)/4 = 0.25 = 1/4.
step 2
we know that ˉx₁ represents the midpoint of the first subinterval 0, 1/4.
the midpoint of 0, 1/4 is
ˉx₁ = (1 + 0)/2 = 1/4 / 2 =
submit skip (you cannot come back)

Explanation:

Step1: Calculate the mid - point of the interval \([0,\frac{1}{4}]\)

The formula for the mid - point of an interval \([a,b]\) is \(\overline{x}=\frac{a + b}{2}\). Here \(a = 0\) and \(b=\frac{1}{4}\).

$$ \overline{x}_1=\frac{0+\frac{1}{4}}{2}=\frac{1}{8} $$

Answer:

\(\frac{1}{8}\)