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. $$\int_{0}^{4} \sqrt{x} d x$$

Short answer

The application of the Trapezoidal rule results in an approximation for the given integral. The prediction based on concavity suggests underestimation of the integral's approximation. The exact integral value calculated matches the prediction of being larger than the approximation.

Step-by-step solution

Apply Trapezoidal Rule

The Trapezoidal Rule approximates the definite integral of the function. According to the rule, with 4 trapezoids (n=4) and limits of integration as 0 and 4, the width of each trapezoid (h) is \((4-0)/4=1\). Compute the Trapezoidal Rule which can be calculated as \(h/2[(f(x_{0}))+2f(x_{1})+2f(x_{2})+2f(x_{3})+f(x_{4})]\), where \(f(x_{i})\) marks the height of each trapezoid at each x value. In this case, the function \(f(x)=\sqrt{x}\)

Use concavity for prediction

Observe that the function \(\sqrt{x}\) is concave upward on the interval [0,4]. In such case, Trapezoidal Rule approximation results in underestimating the area under the curve and thus, the approximation is an underestimate.

Calculate Exact Integral

The original integral, \(\int_{0}^{4}\sqrt{x}dx\), can be calculated directly using the power rule for integration, which states that the integral of \(x^n\) is \(x^{n+1}/(n+1)\). Here \(n=0.5\). Apply the power rule, substitute the limits and calculate the difference to find the exact value of the integral.