Problem 32 - College Algebra and Calculus: An Applied Approach

Question

Use a graphing utility to graph the region bounded by the graphs of the functions. Write the definite integrals that represent the area of the region. (Hint: Multiple integrals may be necessary.) $$ f(x)=x\left(x^{2}-3 x+3\right), g(x)=x^{2} $$

Step-by-Step Solution

Answer

The definite integrals that represent the area of the region bounded by the graphs of the functions are \(|\int_{x_1}^{x_2}(f(x)-g(x))dx|\) and \(|\int_{x_2}^{x_3}(g(x)-f(x))dx|\)

Step 1: Graph the functions

First, we plot the given functions \(f(x) = x(x^2 - 3x + 3)\) and \(g(x) = x^2\) using a graphing tool.

Step 2: Identify intersection points

From the graph, we can find the points at which the two graphs intersect. The intersection points represents the limits of integration for calculating the area bounded by the curves. To find the intersection points, we set the two functions equal to each other and solve for \(x\). \[x(x^2 - 3x + 3) = x^2\] This simplifies to \(x^3 - 3x^2 + 3x = x^2\), and further simplifies to \(x^3 - 4x^2 + 3x = 0\). This equation gives us the x-coordinates of the intersection points, which we denote as \(x_1, x_2\) and \(x_3\).

Step 3: Write the definite integrals

The area \(A\) of the region bounded by the two curves \(f(x)\) and \(g(x)\) from \(x_1\) to \(x_3\) is given by the sum of the absolute values of the definite integrals from \(x_1\) to \(x_2\) and from \(x_2\) to \(x_3\). \[A = |\int_{x_1}^{x_2}(f(x)-g(x))dx| + |\int_{x_2}^{x_3}(g(x)-f(x))dx|\] Either \(f(x)\) is subtracted from \(g(x)\) or vice versa, depending on whether \(f(x)\) or \(g(x)\) is greater on the given interval.