Riemann sums, trapezoids, and signed area

Understand the idea

Approximate a changing quantity by summing rate times interval width. Left, right, and midpoint sums use different sample locations. Trapezoids average endpoint heights; nonuniform partitions require their individual widths. A definite integral is a limit of signed sums, so portions below the axis subtract. For an increasing function, left sums underestimate and right sums overestimate; concavity controls trapezoidal error.

\int_a^b f(x)\,dx=\lim_{n\to\infty}\sum_{i=1}^n f(x_i^*)\Delta x_i,\qquad T=\sum_i\frac{f(x_{i-1})+f(x_i)}2\Delta x_i

Worked example

Given $\displaystyle f(0)=1$ , $\displaystyle f(1)=3$ , and $\displaystyle f(3)=5$ , estimate the integral on $\displaystyle [0,3]$ with trapezoids.

The widths are

\displaystyle 1

and

\displaystyle 2

. The estimate is

\displaystyle \frac12(1+3)(1)+\frac12(3+5)(2)=2+8=10

. Treating both widths as

\displaystyle 1

would be incorrect.

Try it yourself

Find a right Riemann sum for $\displaystyle f(x)=x^2$ on $\displaystyle [0,2]$ using two equal subintervals.

Show solution and reasoning

The width is $\displaystyle 1$ and right endpoints are $\displaystyle 1,2$ . The sum is $\displaystyle 1(1^2+2^2)=5$ .