A kidney-shaped area needs to be determined with the Midpoint Rule, Trapezoidal Rule, and Simpson's Rule. The width is measured at 1 foot interval.

Since the area under a curve equals the definite integral of the curve function, the area can be approximated by using the rule of approximate integration.

The Midpoint Rule approximation is

      Area = Δx(y₀ + y₁ + y₂ + ... + y₁₂)
             = 1(1.5 + 3.5 +4.8 + ... + 1.5)
             = 45.9 ft²

The Trapezoidal Rule approximation is

      Area = Δx/2(y₀ + 2y₁ +2 y₂ + ... + 2y₁₁+ y₁₂)
             = 1/2(1.5 +2(3.5) + 2(4.8) + ... + 2(3.5) + 1.5)
             = 44.4 ft²

3. Simpson's Rule

The Simpson's Rule approximation is

      Area = h/3(y₀ + 4y₁ +2 y₂ + ... + 4y₁₁+ y₁₂)
            = 1/3(1.5 + 4(3.5) + 2(4.3) + ... +4(3.5) + 1.5)
            = 44.73 ft²