Question

In Exercises 1-6, (a) use the Trapezoidal Rule with n = 4 to approximate the value of the integral. (b) Use the concavity of the function to predict whether the approximation is an overestimate or an underestimate. Finally, (c) find the integral's exact value to check your answer. $$\int_{0}^{\pi} \sin x d x$$

Short answer

The approximation of the integral using the Trapezoidal Rule is π[1 + √2 + 2]/4, which is an overestimate. The exact value of the integral \( \int_{0}^{\pi} sin(x) dx \) is 2.

Step-by-step solution

Apply the Trapezoidal Rule

First, the interval [0, π] should be divided into 4 equal parts, is, Δx = (π-0)/4 = π/4. Then, the trapezoidal rule can be applied to approximate the integral, which states \( \int_{a}^{b} f(x) dx \) ≈ Δx/2 [f(x0) + 2f(x1) + 2f(x2) + 2f(x3) + f(x4)] where xi (i from 0 to 4) are the equally spaced points in the interval [0, π].

Approximate the Integral

Substitute values into the Trapezoidal Rule formula yields: Approximation= π/8 [sin(0) + 2sin(π/4) + 2sin(π/2) + 2sin(3π/4) + sin(π)], which simplifies to π[1 + √2 + 2]/4.

Check the Over- or Under-estimate

The function sin(x) is concave down on the interval [0, π/2] and concave up on [π/2, π]. Thus, the Trapezoidal Rule underestimates the true value on [0, π/2] and overestimates on [π/2, π]. As these effects counteract each other, it's uncertain whether the Trapezoidal Rule over- or underestimates without more analysis.

Assess the Exact Value of the Integral

The exact value of the integral \( \int_{0}^{\pi} sin(x) dx \) can be determined by standard calculus method, which equals [-cos(x)] evaluated from 0 to π, and is equal to 2.