How do you calculate the area enclosed between the curves of y = x and y = x^3?

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Calculating the Area Enclosed by the Curves y = x and y = x3

To find the area enclosed between the curves of y = x and y = x3, we first need to determine the points where these two functions intersect. This involves solving for the values of x where:

        x = x3

By rearranging this equation, we have:

        x3 - x = 0

Factoring out x, we find:

        x(x2 - 1) = 0

This produces the solutions:

  • x = 0
  • x = 1
  • x = -1

Now that we have the intersection points, which are x = -1, x = 0, and x = 1, we can set up the integral to calculate the enclosed area.

Since y = x is above y = x3 between these points, we calculate the area by integrating the difference of the functions:

        Area = ∫-11 (x - x3) dx

Now, let’s compute the integral:

        = ∫-11 (x - x3) dx
        = ∫-11 x dx - ∫-11 x3 dx

Calculating these integrals separately:

  1. The integral of x:
    2.         ∫ x dx = (1/2)x2 + C

    Evaluating from -1 to 1:

            = (1/2)(12) - (1/2)(-12) = (1/2) - (1/2) = 0
  2. The integral of x3:
    3.         ∫ x3 dx = (1/4)x4 + C

    Evaluating from -1 to 1:

            = (1/4)(14) - (1/4)(-14) = (1/4) - (1/4) = 0

Substituting back:

    Area = 
    ∫-11 x dx - ∫-11 x3 dx = [0 - 0] = 0

Finally, we must use symmetry, as we know the shape between -1 and 1 is symmetric. The area enclosed can therefore be represented by double the integration from 0 to 1:

    Area = 2 * 
    ∫01 (x - x3) dx
    
    After evaluating:

This results in a final area of:

    Area = 2 * [(1/2)(1) - (1/4)(1)] = 2 * (1/2 - 1/4) = 2 * (1/4) = 1

Therefore, the total area enclosed by the curves is 1 square unit.

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