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Chapter 5: Problem 8

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

Short Answer

Expert verified
The integral \(\int_{-1}^{1} \left[(1-x^{2})-(x^{2}-1)\right] dx = \int_{-1}^{1}\[2-2x^{2}\] dx represents the area under the curve of 2-2x^2 from x = -1 to x = 1. This is an upside-down parabola from -1 to 1 about the x-axis.

Step by step solution

01

- Simplify the integrand

To simplify the integrand, we combine like terms within the brackets. Specifically, we calculate \( (1-x^{2}) - (x^{2} - 1) = 1 - x^{2} - x^{2} + 1 = 2 - 2x^{2}. \) Now our integral is \(\int_{-1}^{1} (2-2x^2) dx \).
02

- Plot the simplified function

The function to plot is \(f(x) = 2 - 2x^2\). This is an upside-down parabola with a vertex at (0,2). Key points are the roots of the function, where f(x) = 0. Set 2 - 2x^2 = 0 to find these: \(x = 卤 sqrt(1)\), so \(x = 卤 1\). Draw the parabola using these points, including the y-intercept at (0,2).
03

- Shade the region

Now that we have the plot, we can shade the region that represents the integral. In this case, the region represents the interval from -1 to 1 under the curve of the function \(2 - 2x^2\). So, we shade the area between the x-axis and our parabola from \(x=-1\) to \(x=1\).

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