Review how we use Riemann sums and the trapezoidal rule to approximate an area under a curve.

What are Riemann sums?

A Riemann sum is an approximation of the area under a curve by dividing it into multiple simple shapes (like rectangles or trapezoids).
In a left Riemann sum, we approximate the area using rectangles (usually of equal width), where the height of each rectangle is equal to the value of the function at the left endpoint of its base.
In a right Riemann sum, the height of each rectangle is equal to the value of the function at the right endpoint of its base.
In a midpoint Riemann sum, the height of each rectangle is equal to the value of the function at the midpoint of its base.
We can also use trapezoids to approximate the area (this is called trapezoidal rule). In this case, each trapezoid touches the curve at both of its top vertices.
For each type of approximation, the more shapes we use, the closer the approximation would be to the actual area.
Resources differ on this point, but we call any approximation that uses rectangles a Riemann sum, and any approximation that uses trapezoids a trapezoidal sum.
Want to learn more about Riemann sums? Check out this video.

Practice set 1: Approximating area using Riemann sums

Problem 1.1
Approximate the area between the xx-axis and f(x)f(x) from x=0x = 0 to x=8x = 8 using a right Riemann sum with 33 unequal subdivisions.
xx00334488
f(x)f(x)2255771111
The approximate area is
  • Your answer should be
  • an integer, like 66
  • a simplified proper fraction, like 3/53/5
  • a simplified improper fraction, like 7/47/4
  • a mixed number, like 1 3/41\ 3/4
  • an exact decimal, like 0.750.75
  • a multiple of pi, like 12 pi12\ \text{pi} or 2/3 pi2/3\ \text{pi}
units2^2.

Want to try more problems like this? Check out this exercise.

Practice set 2: Approximating area using the trapezoidal rule

Problem 2.1
Approximate the area between the xx-axis and h(x)h(x) from x=3x = 3 to x=11x = 11 using a trapezoidal sum with 44 equal subdivisions.
xx335577991111
h(x)h(x)336644881212
The approximate area is
  • Your answer should be
  • an integer, like 66
  • a simplified proper fraction, like 3/53/5
  • a simplified improper fraction, like 7/47/4
  • a mixed number, like 1 3/41\ 3/4
  • an exact decimal, like 0.750.75
  • a multiple of pi, like 12 pi12\ \text{pi} or 2/3 pi2/3\ \text{pi}
units2^2.

Want to try more problems like this? Check out this exercise.
Loading