Question

The integrand of the definite integral is a difference of two functions. Sketch the graph of each function and shade the region whose area is represented by the integral. $$ \int_{2}^{3}\left[\left(\frac{x^{3}}{3}-x\right)-\frac{x}{3}\right] d x $$

Short answer

After graphing, the shaded region starts from \(x = 2\), up to \(x = 3\), below the curve \(f(x) = \frac{x^{3}}{3}\) and above the curve \(g(x) = \frac{4x}{3}\). This area represents the definite integral.

Step-by-step solution

Simplify the Integral Expression

Simplify the integral expression by combining like terms: \( \int_{2}^{3}\left[\left(\frac{x^{3}}{3}-x\right)-\frac{x}{3}\right] d x = \int_{2}^{3}\left[\frac{x^{3}}{3}-x-\frac{x}{3}\right] d x = \int_{2}^{3}\left[\frac{x^{3}}{3}-\frac{4x}{3}\right] d x \).

Break down the simplified Expression into two Function Graphs

We have two functions based on the given expression: \(f(x)= \frac{x^{3}}{3}\) and \(g(x)=\frac{4x}{3}\). Plot both \(f(x)\) and \(g(x)\) on the same graph.

Shade the Region

The area given by the definite integral is the area between these two curves. Shade this region on the plot. The area begins at \(x=2\) and ends at \(x=3\) under the function \(f(x)-g(x)\).