Question

Use the Midpoint Rule with \(n=4\) to approximate the area of the region. Compare your result with the exact area obtained with a definite integral. $$ f(x)=\sqrt{x}, \quad[0,1] $$

Short answer

The Midpoint Rule provides an estimated value whereas a definite integral provides the exact value. The further division into more intervals will give a result closer to the exact integral.

Step-by-step solution

Applying the Midpoint Rule

The Midpoint Rule formula states that the integral of \(f\) over the interval \([a, b]\) can be approximated as: \((b - a) \cdot f((a + b)/2)\). Since \(n = 4\), we divide the interval \([0, 1]\) into 4 equal parts, which gives us the midpoints of 0.125, 0.375, 0.625 and 0.875. Now all we need to do is plug these values into our function and sum up the results: \(0.25 * (\sqrt{0.125} + \sqrt{0.375} + \sqrt{0.625} + \sqrt{0.875})\).

Calculating the integral

The function \(f(x) = \sqrt{x}\) has antiderivative \(F(x) = \frac{2}{3}x^{1.5}\). The definite integral over the interval \([0, 1]\) can be calculated with the Fundamental Theorem of Calculus: \(F(1) - F(0) = \dfrac{2}{3}(1^{1.5}) - \dfrac{2}{3}(0^{1.5}) = \dfrac{2}{3}\).

Comparison

Now you have the two areas: the estimated area using Midpoint Rule and the exact area calculated using the definite integral. Compare the two results to understand the efficiency of the Midpoint Rule.