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. $$\int_{0}^{2} x^{2} d x$$

Short answer

First, using Trapezoidal Rule we get an approximate value for the integral. The concavity of the function tells us that this is an underestimate of the actual integral. Finally, using integral calculus, we find the exact value of the integral, which verifies the approximation.

Step-by-step solution

Trapezoidal Approximation

The formula for the Trapezoidal Rule is \[ \int_{a}^{b} f(x) dx \approx \frac{b - a}{2n} \left[f(x_0) + 2f(x_1) + 2f(x_2) + \ldots + 2f(x_{n-1}) + f(x_n)\right] \]To apply this to our function \(x^2\) from 0 to 2 with \(n = 4\), we need to calculate \(f(x)\) for \(x_0=0\), \(x_1=0.5\), \(x_2=1.0\), \(x_3=1.5\), and \(x_4=2.0\). We will then use these to calculate the Trapezoidal Rule approximation of the integral.

Predict Overestimate or Underestimate

The concavity of \(x^2\) from 0 to 2 on the x-axis is upwards, which means the Trapezoidal Rule will always produce an underestimate of the integral because it places straight line segments above the curve.

Exact Integral Calculation

To calculate the exact value of the integral, we use the Power Rule of integral, which states\[ \int_{a}^{b} x^n dx = \frac{1}{n+1} b^{n+1} - \frac{1}{n+1} a^{n+1} \]For our problem, the exact integral of \(x^2\) from 0 to 2 would be calculated as\[\frac{1}{2+1} [2^{2+1} - 0^{2+1}] = \frac{2^{3}}{3} - \frac{0^{3}}{3}.\]