Problem 6 - Essential Calculus

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. $$ \int_{2}^{3}\left[\left(\frac{x^{3}}{3}-x\right)-\frac{x}{3}\right] d x $$

Step-by-Step Solution

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 1: Simplify the Integral Expression

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

Step 2: 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.

Step 3: 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 \(f(x)-g(x)\).