Calculating the Area Under the Curve y x2 - 2x Between x 0 and x 2

The equation y x^2 - 2x represents a parabola that opens upwards, with its vertex at the point (1, -1). This means that, for the interval from x 0 to x 2, the curve lies below the x-axis except at the points where it intersects the x-axis (i.e., at x 0 and x 2). Therefore, to find the area under this curve, we need to integrate the function between these points, taking into account that the result will be negative since the graph is below the x-axis.

Definite Integral Calculation

To find the area under the curve, we evaluate the definite integral of the function y x^2 - 2x from x 0 to x 2.

The integral is given by:

(A int_0^2 (x^2 - 2x), dx)

Evaluating the integral:

First, we find the antiderivative of the function x^2 - 2x:

(int (x^2 - 2x), dx frac{x^3}{3} - x^2 C)

Next, we apply the fundamental theorem of calculus by evaluating this antiderivative at the bounds 0 and 2:

[A left[ frac{x^3}{3} - x^2 right]_0^2 left( frac{2^3}{3} - 2^2 right) - left( frac{0^3}{3} - 0^2 right)]

This simplifies to:

[A left( frac{8}{3} - 4 right) - 0 frac{8}{3} - frac{12}{3} frac{-4}{3}]

Since the area is always positive, we take the absolute value of the result:

(A frac{4}{3})

Graphical Representation and Area Calculation

To further understand the calculation, we can graph the curve y x^2 - 2x between x 0 and x 2. By plotting the points (0, 0), (1, -1), and (2, 0), we can see that the curve creates a triangular region below the x-axis. This region can be further divided into two smaller regions (blue and green) for more precise calculations.

The blue area is given by:

(A_{text{blue}} int_2^3 (x^2 - 2x), dx)

The green area, taking into consideration the negative integral, is given by:

(A_{text{green}} -int_2^0 (x^2 - 2x), dx int_0^2 (x^2 - 2x), dx)

Both areas will sum up to the total area under the curve:

A_{text{total}} frac{4}{3} frac{4}{3} frac{8}{3}

It is important to note that these calculations assume correct boundaries and the correct application of the definite integral. Misinterpretations or incorrect boundaries can lead to different results, such as the area being reported as 19/3 by some, which likely arose from a misinterpretation of the problem or calculation error.

To summarize, the area under the curve y x^2 - 2x between x 0 and x 2, taking into account that the curve lies below the x-axis, is given by the definite integral and can be calculated as (frac{4}{3}).