How to Integrate Over the Perimeter of a Square

Isaac Newton developed integration in order to explain his gravitational theory. An integral, in essence, is a sum that operates over an infinite number of terms; each term tends to be extremely small, close to zero. By using algebra and calculus, integrals allow the user to calculate areas and volumes from regular and irregular shaped objects. Calculating the area of a square, by integrating data from the perimeter, is a basic calculus exercise that will reinforce the topics of integration, exponents and functions.

Instructions

    • 1

      Divide the perimeter of the square by four to get the length of a side of the square. Assign a variable to this quantity.

      For example:

      Perimeter of the square = P

      P / 4 = L (side of the square)

    • 2

      Draw an XY axis. Create a square shape with side equal to L.

      To create the square shape, create a function:

      Y = L

      This function will be the top of the square. The bottom will be the "X" axis. The left side will be the "Y" axis.

      The right side will be the vertical line X = L

    • 3

      Set up an integral for the function "Y = L" from zero to "L" using "X" as the variable.

      The integral will be:

      Integral ( L ) from [ 0, L ]

    • 4

      Solve the integral. You can use the online integrator (See Resources).

      After solving the integral you will get:

      ( L ) x ( X ) evaluated from [ 0, L ] =

      ( L ) x ( L - 0 ) = L^2

    • 5

      Replace the value of L in terms of the perimeter of the square (from Step 1).

      Since P / 4 = L

      L^2 = ( P / 4 )^2

      L^2 = P^2 / 16

      The area of the square will be: P^2 / 16

      Where P is the perimeter.

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