Essential Knowledge 3.2A3

Essential Knowledge 3.2A3 Students will know that the information in a definite integral can be translated into the limit of a related Riemann sum, and the limit of a Riemann sum can be written as a definite integral.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Once again, our statement from last Essential Knowledge was

, where and x_i^* is a value in the ith subinterval.

Not gonna spend too much time on this Essential Knowledge. To convert between integral notation and summation notation, just look at where all the parts go.

• a and b are obvious

• f(x) is the same thing as f(x*_i)

• dx is Δx_i

• integral notation doesn't include n because integrals always have n → ∞. Also, integrals don't need to worry about (b - a)/n in integral notation because integrals just use the special rules that are the opposite of the derivative rules, rather than solving a limit the old fashioned way.

  • (With the Riemann sum, you can actually solve for area using special summation rules, and thus never even need to know anything about derivatives. We don't do all that extra work, thankfully, because I can tell you it's no fun. Hooray for using the inverse of derivative rules!)