Question

Evaluate. Then interpret the result in terms of the area above and/or below the x-axis. | (x3 - 2x) dx 2 [ = ] (x3 - 2x) dx = 1 (Type an integer or a simplified fraction.) 2

Answer 1

We evaluate the integral using the Fundamental theorem of Calculus, by using and antiderivative of the funtion in the integrand;

An antiderivative of x^3 - 2 x is = 1/4 x^4 - x^2

So we evaluate this antiderivative at the two limits of integration:

At x = 1 the antiderivative becomes: 1/4 (1) - 1 = - 3/4

At x = - 1/2 the antiderivative becomes: 1/4 (-1/2)^4 - (-1/2)^2 = - 15/64

Now we subtract the evaluation at the upper limit minus the evaluation at the lower limit:

- 3/4 - (-15/64) = - 33/64

Allow me to show you the actual area we have calculated in a graph for the integration:

The curve in blue is the original function you provided : f(x) = x^3 - 2x

You can see that there is an area above the x axis that has been integrated and that gives as a result a positive number.

The area below the x axis is the part of the integral that provides the negative part which as you see is dominant in this calculation, therefore resulting in a negative final result.

Please, make sure you type -33/64 in the box provided.