Integration of a Constant Function

Suppose wire costs 3 cents per centimeter. This constant rate can be expressed as y = f(x) = 3 for all values of x, where x is the length of wire. The area from 0 to 1 represents the 3 cents that the first centimeter of wire costs. The area from 1 to 2 represents the 3 cents that the second centimeter of wire costs. All of the area under the “curve” from 0 to 5 represents the total, 3 + 3 + 3 + 3 + 3, that the first five centimeters of wire cost, 15 cents. Adding the area under the curve is represented by the integral symbol ∫. The integral of the function y = 3 is 3x. To evaluate this integral between the points x=0 and x=5, subtract the value of 3x at x=0 from the value of 3x at x=5. The total cost is 15 - 0, or 15 cents.