Trapezoidal rule

Definition

The trapezoidal rule is a numerical method used to estimate the definite integral of a function by approximating the area under its curve with trapezoids. This method connects closely to Riemann sums, as it essentially refines the idea of summing up areas by using linear approximations to better capture the shape of the curve, making it more accurate than simple rectangles.

5 Must Know Facts For Your Next Test

1. The trapezoidal rule improves accuracy by approximating the curve with straight line segments rather than using rectangles, which can lead to underestimations or overestimations.
2. To apply the trapezoidal rule, you divide the interval into 'n' equal subintervals and calculate the area of each trapezoid formed by the function's values at the endpoints.
3. The formula for the trapezoidal rule is given by $$ T_n = \frac{b-a}{2n} \left( f(a) + 2\sum_{i=1}^{n-1} f(x_i) + f(b) \right) $$, where $$ [a, b] $$ is the interval of integration and $$ x_i $$ are the points dividing that interval.
4. The trapezoidal rule has a convergence rate that depends on the smoothness of the function being integrated; smoother functions yield more accurate results.
5. While the trapezoidal rule can be used for any function, it is particularly effective when integrating functions that are approximately linear over small intervals.

Review Questions

• How does the trapezoidal rule enhance the accuracy of approximating integrals compared to Riemann sums?
  • The trapezoidal rule enhances accuracy by replacing rectangles used in Riemann sums with trapezoids, which better approximate the area under a curve. While Riemann sums can lead to errors based on whether they overestimate or underestimate areas due to their rectangular shape, trapezoids adjust for curvature, making it easier to capture more precise areas. By averaging function values at each endpoint of an interval and using straight lines to connect these points, the trapezoidal rule provides a more reliable estimate of integrals.
• Discuss how you would implement the trapezoidal rule for a given function over a specific interval.
  • To implement the trapezoidal rule for a given function over an interval, you start by determining your interval boundaries, say [a, b]. Next, choose how many subintervals 'n' you want to divide this interval into. Calculate the width of each subinterval and then evaluate the function at each endpoint as well as at intermediate points. Finally, apply the trapezoidal formula to sum up all trapezoidal areas based on these evaluations to obtain an approximation of the integral.
• Evaluate how changing the number of subintervals 'n' affects the accuracy of the trapezoidal rule in numerical integration.
  • Increasing the number of subintervals 'n' in the trapezoidal rule generally leads to greater accuracy in approximating integrals. This is because more subintervals allow for smaller widths, reducing the error associated with each individual trapezoid's approximation of curvature. However, there's a trade-off in computational effort; while accuracy improves with more subdivisions, it also requires more calculations. Ultimately, finding an optimal balance between 'n' and computational efficiency is key for precise numerical integration.

Related terms

Riemann Sum:

A Riemann sum is an approximation of the integral of a function by dividing the area under the curve into rectangles and summing their areas.

Definite Integral:

A definite integral calculates the net area between a function and the x-axis over a specified interval, providing a precise measure of accumulation.

Numerical Integration: Numerical integration refers to a range of algorithms for calculating the value of integrals, especially when an analytical solution is difficult or impossible to obtain.