Problem 13 - Essential Calculus

Question

Evaluate the definite integral of the algebraic function. Use a graphing utility to verify your result. $$ \int_{0}^{1} \frac{x-\sqrt{x}}{3} d x $$

Step-by-Step Solution

Answer

The value of the definite integral of the function \(\frac{x-\sqrt{x}}{3}\) from 0 to 1 is -1/2.

Step 1: Break Down the Function

First, it's important to see the function \(\frac{x-\sqrt{x}}{3}\) as the sum of two different functions divided by 3: \(\frac{x}{3}-\frac{\sqrt{x}}{3}\). This will make the integration easier since each function can be integrated separately.

Step 2: Apply the Power Rule to each Function

Applying the power rule of integration to \(x/3\) we get \(\frac{x^2}{6}\) and to \(\sqrt{x}/3=\frac{x^{1/2}}{3}\) we get \(\frac{2}{3}x^{3/2}\). Remember, the power rule of integration states that the integral of \(x^n\) with respect to \(x\) is \(1/(n+1)*x^{n+1}\). Then, the definite integral from 0 to 1 of the function is obtained by evaluating the sum of these two antiderivatives at 1 and then subtracting the result of evaluating them at 0.

Step 3: Evaluate the Definite Integral

The definite integral from 0 to 1 is obtained by evaluating \( (\frac{x^2}{6}-\frac{2}{3}x^{3/2})\) at x=1, which gives \( \frac{1}{6}-\frac{2}{3}\) and at x=0, which gives 0 (since any number multiplied by 0 is 0). The result is then \((\frac{1}{6}-\frac{2}{3}) - 0 = -\frac{1}{2}\).

Step 4: Verify the Result with a Graphing Utility

As a final check, one can use a graphing utility to plot the function \(\frac{x-\sqrt{x}}{3}\) and verify that the area under the curve from 0 to 1 is indeed equal to -1/2. It should be noted that negative area corresponds to the portion of the graph that is below the x-axis.