In some situations, it is impossible to find the exact value of a definite integral. For these cases, an approximate value of definite integrals are needed. In this section, several rules to find the approximate value of a definite integral are introduced.

Midpoint Rule

Since the definite integral is defined as a limit of Riemann sum, any Riemann sum could be used as an approximation to the integral. Recall, a definite integral is

where f(x) is a continuous function on interval [a, b]. To apply an approximation, first divide the interval [a, b] into n subintervals of equal length

     Δx = (b - a)/n

Next, choose a x in each subinterval and the integration can be approximated as

where x_i^* is any point in the ith subinterval [x_i-1,x_i].

If x_i^* is choose to be the midpoint of each subinterval, this approximation is called the Midpoint Rule.

     x_i^* = 1/2(x_i + x_i+1)

The definite integral of function f(x) in an interval [a, b] equals the area bounded by curve y = f(x), x = a, x = y, and y = 0. Therefore, Midpoint Rule actually is using a rectangle to approximate the area under the curve over a subinterval [x_i-1 , x_i].

Trapezoidal Rule

When using the Riemann sum to approximate definite integration, the approximation is called left endpoint approximation if the left point is choose as x_i^*. In the same way, the approximation is called right endpoint approximation if the right point is choose as x_i^* . The Trapezoidal Rule is from averaging these two approximation.  

    

Actually the Trapezoidal Rule is using a trapezoid to approximate the area over a subinterval. For example, in the interval [x₀, x₁], the area of the trapezoid is

      ΔA = Δx/2[f(x₀) + f(x₁)]

When all the trapezoids at each interval are added together, the total gives,

     A = Δx/2[(f(x₀) + f(x₁)) + (f(x₁) + f(x₂) + ...
            + (f(x_n-1) + f(x_n))]
        = Δx/2 + [f(x₀) + 2f(x₁) + ... + 2f(x_n-1) + f(x_n)]

which is the Trapezoidal Rule approximation.

Simpson's Rule

Simpson's Rule on Interval [-h, h]

Simpson's Rule for approximation integration uses parabolas to approximate the curved edge. First divide interval [a, b] into n equal length subintervals

     h = Δx = (b - a)/n

where n is an even number.

On each consecutive pair of intervals, the curve y = f(x) can be approximated by a parabola y = Ax² + Bx + C. To better understand this, consider a subinterval between x₀ and x₂. To simply the calculation, assume

       x₀ = -h
       x₁ = x₀ + h = 0
       x₂ = x₀ + 2h = h

Then the area under the parabola is

Since point P₀, P₁ , and P₂ are on both curves,

      y₀ = f(x₀) = A(-h)² - Bh + C
      y₁ = f(x₁) = C
      y₂ = f(x₂) = A(h)² + Bh + C

Adding the above three equations together gives

      y₀ + y₁ + y₂ = 2Ah² + 3C

Therefore, the area under the parabola can be rewritten as

      A = h/3( 2Ah² + 6C)
         = h/3(y₀ + y₁ + y₂ + 3y₁)
         = h/3(y₀ + 4y₁ + y₂)

Shifting the parabola between [-h, h] to [x₀, x₂] does not change the area under it. Similarly, the area under the parabolas through point P₂, P₃, and P₄ are

       A = h/3(y₂ + 4y₃ + y₄)

Calculating the area under all the parabolas and adding them together gives

If n = 6, the integration can be approximated as

Note the pattern of coefficients is: 1, 2, 4, 2, 4, 2, ..., 4, 2, 4, 1.