Trapezoidal, midpoint and Simpson's rules practice problems
The trapezoidal, midpoint and Simpson's rules, and how close they come. 10 problems with worked solutions; in 10 of them every equation is proved by a computer algebra system.
Approximate \( \displaystyle \int_{0}^{2} x^{2}\, dx \) using the Simpson's rule with \( \displaystyle n = 6 \).
Approximate \( \displaystyle \int_{0}^{1} \frac{4}{x^{2} + 1}\, dx \) using the midpoint rule with \( \displaystyle n = 4 \).
Approximate \( \displaystyle \int_{0}^{1} \frac{1}{x^{2} + 1}\, dx \) using the midpoint rule with \( \displaystyle n = 5 \).
Approximate \( \displaystyle \int_{1}^{3} \frac{1}{x}\, dx \) using the trapezoidal rule with \( \displaystyle n = 4 \).
Approximate \( \displaystyle \int_{0}^{2} x^{3} + 1\, dx \) using the trapezoidal rule with \( \displaystyle n = 5 \).
Approximate \( \displaystyle \int_{1}^{2} \frac{1}{x^{2}}\, dx \) using the Simpson's rule with \( \displaystyle n = 6 \).
Approximate \( \displaystyle \int_{0}^{2} x^{2}\, dx \) using the trapezoidal rule with \( \displaystyle n = 4 \).
Approximate \( \displaystyle \int_{1}^{3} \frac{1}{x}\, dx \) using the Simpson's rule with \( \displaystyle n = 6 \).
Approximate \( \displaystyle \int_{0}^{2} \frac{x}{x + 1}\, dx \) using the Simpson's rule with \( \displaystyle n = 6 \).
Approximate \( \displaystyle \int_{0}^{2} \frac{x}{x + 1}\, dx \) using the midpoint rule with \( \displaystyle n = 5 \).