A+ » VCE » Maths Methods U3 & 4 Master Notes » A3 Calculus » Sum of Area

Sum of Area

7.2 Sum of Area and Definite Integrals

Definite Integrals as an Accurate Estimate

  • As the number of rectangles increases, we are able to obtain more and more accurate estimations of the rectangle. Ultimately, when we split the area into infinitely many slim rectangles, the difference between the rectangle’s horizontal edge and the curve’s curvy line in the infinitely small range of x is not noticeable, and so we can get an accurate estimate.
  • In fact, we call the area under the graph of y=f(x) from x=a to x=b as the definite integral of f(x) from x=a to x=b. It is written as

\int_{a}^{b} f(x) d x

where b>a.

  • The function f is called the integrand, and a and b are the lower and upper limits of the integral.
  • Using the summation notation, it can be rewritten as
Read More »7.2 Sum of Area and Definite Integrals

7.1 Estimating Area as Sum of Rectangles

Estimating Area as Sum of Rectangles

  • The area under a graph within an interval x \in [a,\ b] can be estimated by a sum of rectangles for smaller intervals of x.
  • There are two types of constructing these rectangles, namely the left-endpoint estimate and right-endpoint estimate. Both estimates will probably lead to a different answer.
  • There is no rule as left-endpoint estimate will yield a smaller or bigger area. It depends on the curve, but a general rule can be applied:

i) For f decreasing over [a,\ b]: left-endpoint estimate ≥ true area ≥ right-endpoint estimate

ii) For f increasing over [a,\ b]: left-endpoint estimate ≤ true area ≤ right-endpoint estimate

  • The steps of estimating area under a curve as sum of rectangles are:
Read More »7.1 Estimating Area as Sum of Rectangles