Final answer:
To approximate the given integral, you can use the Trapezoidal Rule, Midpoint Rule, or Simpson's Rule. Each method involves dividing the interval into subintervals and approximating the integral by summing specific shapes formed by the function values at those intervals. The approximate integral value is obtained by multiplying the sum by the width of one subinterval.
Step-by-step explanation:
To approximate the given integral, you can use three different numerical integration methods: Trapezoidal Rule, Midpoint Rule, and Simpson's Rule. Let's go through each method:
Trapezoidal Rule:
- Divide the interval into 'n' subintervals of equal width.
- Approximate the integral by summing the areas of trapezoids formed by each pair of adjacent points.
- Multiply the sum by the width of one subinterval to find the approximate integral value.
Midpoint Rule:
- Divide the interval into 'n' subintervals of equal width.
- Approximate the integral by summing the areas of rectangles formed by the midpoints of each subinterval and the function values at those midpoints.
- Multiply the sum by the width of one subinterval to find the approximate integral value.
Simpson's Rule:
- Divide the interval into 'n' subintervals of equal width.
- Approximate the integral by summing the areas of parabolic segments formed by three consecutive points.
- Multiply the sum by the width of one subinterval to find the approximate integral value.