Problem 8 - College Algebra and Calculus: An Applied Approach

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. $$ \int_{-1}^{1}\left[\left(1-x^{2}\right)-\left(x^{2}-1\right)\right] d x $$

Step-by-Step Solution

Answer

The area represented by the integral is 4 square units.

Step 1: Graph the functions

Firstly, the individual functions \(1-x^{2}\) and \(x^{2}-1\) need to be graphed over the interval \([-1, 1]\). \(1-x^{2}\) is a downward-facing parabola with its vertex at (0,1). On the other hand, \(x^{2}-1\) is an upward-facing parabola with its vertex at (0,-1). Overlap these two sketches on the same graph.

Step 2: Identify the area under the functions

Now, you need to identify the area between these two functions, which is essentially the region bounded by these two parabolas from \(-1\) to \(1\). This region should be shaded on the graph.

Step 3: Evaluate the integral

Next, evaluate the definite integral \(\int_{-1}^{1} [(1-x^{2})-(x^{2}-1)] dx\). To do this, first simplify the integrand resulting in \(\int_{-1}^{1} (2-2x^{2}) dx\). Now evaluate this simplified integral, which can be broken down as the sum of integrals as follows: \(\int_{-1}^{1}2dx - \int_{-1}^{1}2x^{2}dx\). Evaluate these two separate integrals and sum up their results.

Step 4: Interpret the result

The result of the definite integral will represent the area of the shaded region in the graph. The sign of the result will also provide information about the position of the shaded area with respect to the x-axis.