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. $$ \int_{-2}^{2}\left[2 x^{2}-\left(x^{4}-2 x^{2}\right)\right] d x $$

Short answer

The area under the curve between -2 and 2 is represented by the definite integral of the function \(4x^{2} - x^{4}\).

Step-by-step solution

Simplify the function

In order to simplify the calculation, the function inside the integral can be simplified. It is given by \[ 2x^{2} - (x^{4} - 2x^{2}) \]. This can be simplified to \[ 2x^{2} - x^{4} + 2x^{2} = 4x^{2} - x^{4} \]

Sketch the function

The function \(4x^{2} - x^{4}\) is of even degree, so it will have a minimum value and will open downwards. This function is known as a well-known parabola. We apply symmetry to our sketch along the y-axis. This function intersects the x-axis at -2, 0, and 2.

Evaluate the integral

To represent the area under the curve, we must perform the definite integral of our function from -2 to 2. This is done by \[ \int_{-2}^{2} (4x^{2} - x^{4}) dx \]. Evaluating this integral gives us the shaded region or the area under the curve between -2 and 2.