Problem 16 - College Algebra and Calculus: An Applied Approach

Question

Sketch the region bounded by the graphs of the functions and find the area of the region. $$ y=x^{3}-2 x+1, y=-2 x, x=1 $$

Step-by-Step Solution

Answer

The area is computed by the previously mentioned integration formula with the found limits of integration.

Step 1: Graphing the function

First, the functions \(y=x^{3}-2x+1\), \(y=-2x\), and \(x=1\) must be graphed. This can be done using graph paper or a graphing software program. After drawing the graph, it's time to find the region bounded by the three curves.

Step 2: Detect the bounded region

Once the graph of the function is drawn, the next step is to identify the bounded region. This is the region enclosed by all three functions. As observed from the graph, the bounded region will be between the graphs of \(y=x^{3}-2x+1\), \(y=-2x\), and the vertical line \(x=1\). Notice that the intersection points of the curves are the limits of integration.

Step 3: Find the limits of integration

The intersection points of the curves are essentially the limits for the definite integral to calculate the area. Given that the 2 functions intersect when \(x^{3}-2x+1=-2x\), solving this equation reveals the two points which will be used as the limits of integral. The intersection point with the line \(x=1\) also need to be identified.

Step 4: Compute the area

Now that the limits of integration are known, the area can be computed. The area of a region bounded by two curves \(f(x)\) and \(g(x)\) is given by the formula \(Area=\int_{a}^{b} |f(x)-g(x)| dx\). Here, \(f(x)\) and \(g(x)\) refer to \(y=x^{3}-2x+1\) and \(y=-2x\) respectievly, and 'a' and 'b' are the points of intersection.