Evaluating and Interpreting the Integral of x^3 - 2x

In this article, we will explore the integral of the function x^3 - 2x from the lower limit of -1/2 to the upper limit of 1. By evaluating this integral, we can determine the area under the curve of the function, as well as the area above and below the x-axis.

The integral of a function represents the accumulation of the function's values over a given interval. In this case, we are interested in the integral of the function x^3 - 2x from -1/2 to 1.

The formula for this integral is:

∫_{-1/2}^{1} (x^3 - 2x) dx

To evaluate this integral, we can use the fundamental theorem of calculus, which states that the integral of a function is the antiderivative of that function, evaluated at the given limits.

The antiderivative of x^3 - 2x is:

∫ (x^3 - 2x) dx = (x^4 / 4) - 2x^2 / 2 + C

where C is the constant of integration.

Substituting the limits, we get:

∫_{-1/2}^{1} (x^3 - 2x) dx = [(1^4 / 4) - (2 * 1^2 / 2)] - [(-1/2)^4 / 4 - (2 * (-1/2)^2 / 2)]

Simplifying the expression, we get:

∫_{-1/2}^{1} (x^3 - 2x) dx = (1 - 1) - ((-1/16) - 1/8)

Evaluating the result, we get:

∫_{-1/2}^{1} (x^3 - 2x) dx = 0 - (-1/16 - 1/8)

Simplifying further, we get:

∫_{-1/2}^{1} (x^3 - 2x) dx = 3/16

The result of the integral is 3/16, which represents the net area under the curve of the function x^3 - 2x from -1/2 to 1.

To interpret the result, we need to consider the sign of the function. The function x^3 - 2x is positive for x > 2/3 and negative for x < 2/3. This means that the area under the curve is positive for x > 2/3 and negative for x < 2/3.

The net area of 3/16 represents the difference between the positive area above the x-axis and the negative area below the x-axis. This means that the positive area above the x-axis is larger than the negative area below the x-axis.

In conclusion, the integral of x^3 - 2x from -1/2 to 1 is 3/16, which represents the net area under the curve of the function. The positive area above the x-axis is larger than the negative area below the x-axis, resulting in a positive net area.