Unit 8: Numerical Integration
When there's no antiderivative to fall back on, straight lines and parabolas estimate the area instead.
4 topics · 17 questions
Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.
The Trapezoidal Rule
- How it worksHow does averaging the left and right Riemann sums produce exactly the trapezoidal rule's formula?
- Common mistakeWhy doesn't the trapezoidal rule give the exact area even though trapezoids fit a curve better than rectangles?
- Worked exampleFind the trapezoidal rule estimate for the integral from 0 to 4 of x2 dx using 4 subintervals.
- Going furtherWhat happens to the trapezoidal rule's accuracy as the number of subintervals grows very large?
Your turn1 of 2
Find the trapezoidal rule estimate for the integral from 0 to 5 of 4x2 dx using 4 subintervals.Simpson's Rule
- How it worksHow does Simpson's rule weight the function values 1, 4, 2, 4, and on to fit a parabola through each triple of points?
- Worked exampleFind the Simpson's rule estimate for the integral from 0 to 4 of x2 dx using 4 subintervals.
- Worked exampleFind the Simpson's rule estimate for the integral from 0 to 2 of the exponential ex dx using 4 subintervals.
- Common mistakeWhy doesn't Simpson's rule work when the number of subintervals is odd?
Your turn1 of 2
Find the Simpson's rule estimate for the integral from 0 to 2 of 3ex dx using 4 subintervals.The Error Bound for the Trapezoidal Rule
- How it worksHow does the maximum size of the second derivative on an interval control the trapezoidal rule's error bound?
- Going furtherWhat happens to the trapezoidal error bound as the number of subintervals doubles?
- Worked exampleFind the error bound for the trapezoidal rule estimate of the integral from 0 to 4 of x2 dx with 4 subintervals.
- Common mistakeWhy doesn't a smaller number of subintervals always mean a proportionally smaller error bound?
Your turn1 of 2
Find the error bound for the trapezoidal rule estimate of the integral from 0 to 5 of 4x2 dx with 4 subintervals.The Error Bound for Simpson's Rule
- How it worksHow does the maximum size of the fourth derivative, instead of the second, control Simpson's rule's error bound?
- Worked exampleFind the error bound for the Simpson's rule estimate of the integral from 0 to 2 of ex dx with 4 subintervals.
- Common mistakeWhy doesn't Simpson's rule need more subintervals than the trapezoidal rule to hit the same accuracy?
- Going furtherWhat happens to Simpson's rule's error bound for a function whose fourth derivative is exactly zero?
- Ties togetherHow does comparing the trapezoidal and Simpson error bounds explain why Simpson's rule is the default in practice?
Your turn1 of 2
Find the error bound for the Simpson's rule estimate of the integral from 0 to 2 of 3ex dx with 4 subintervals.