Approximate Integration (including Error Bounds)
MAT1242: Calculus II
University of Eswatini.
Approximate integration, also known as numerical integration, is a technique used to estimate the
value of a definite integral when an exact analytical solution is not feasible. It involves dividing the
interval of integration into smaller subintervals and approximating the integral using various numerical
methods.
1. Rectangular Rule (Midpoint Rule)
The rectangular rule approximates the integral by evaluating the function at the midpoint of each
subinterval and multiplying it by the width of the subinterval. The formula for the midpoint rule is:
where:
● (n) is the number of subintervals.
● is the midpoint of the (i)-th subinterval.
● is the width of each subinterval.
Example:
Let’s approximate using the midpoint rule with (n = 4).
1. Divide the interval into 4 subintervals: .
2. Evaluate the function at the midpoints:
○
○
○
○
3. Sum up the products: .
2. Trapezoidal Rule
MAT1242: Calculus II
University of Eswatini.
Approximate integration, also known as numerical integration, is a technique used to estimate the
value of a definite integral when an exact analytical solution is not feasible. It involves dividing the
interval of integration into smaller subintervals and approximating the integral using various numerical
methods.
1. Rectangular Rule (Midpoint Rule)
The rectangular rule approximates the integral by evaluating the function at the midpoint of each
subinterval and multiplying it by the width of the subinterval. The formula for the midpoint rule is:
where:
● (n) is the number of subintervals.
● is the midpoint of the (i)-th subinterval.
● is the width of each subinterval.
Example:
Let’s approximate using the midpoint rule with (n = 4).
1. Divide the interval into 4 subintervals: .
2. Evaluate the function at the midpoints:
○
○
○
○
3. Sum up the products: .
2. Trapezoidal Rule
