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Chapter 11: Problem 2

Use a graphing utility to graph the integrand. Use the graph to determine whether the definite integral is positive, negative, or zero. $$ \int_{-2}^{2} x \sqrt{x^{2}+1} d x $$

Short Answer

Expert verified
The definite integral \( \int_{-2}^{2} x \sqrt{x^{2}+1} d x \) is zero.

Step by step solution

01

Determine the function to be Integrated

The function that is being integrated is \( f(x) = x \sqrt{x^{2}+1} \). This is the function that will be graphed.
02

Graph the Function

Plot the function \( f(x) = x \sqrt{x^{2}+1} \) in a graphing utility. Pay special attention to the range of the x-values. Here, we are interested in the interval from -2 to 2.
03

Examine the graph

Examine the graph over the interval from -2 to 2. If the graph lies above the x-axis in this interval, then the definite integral will be positive. If it lies below the x-axis, then the definite integral will be negative. If it's exactly on the x-axis, the integral will be zero. In this case, one can observe that the graph lies both above and below the x-axis, with the area below the x-axis matching the area above, indicating that the overall integral will be zero.

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Most popular questions from this chapter

Sketch the region bounded by the graphs of the functions and find the area of the region. $$ f(x)=\sqrt[3]{x}, g(x)=x $$

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.) $$ f(x)=x\left(x^{2}-3 x+3\right), g(x)=x^{2} $$

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