SUNY Geneseo Department of Mathematics

Definite Integrals

Friday, November 3 - Monday, November 6

Math 221 05
Fall 2017
Prof. Doug Baldwin

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Previous Lecture

Questions?

“Cylinder in Sphere”? What does that problem ask in problem set 8?

If you imagine putting a cylinder inside a sphere, the size and shape of the cylinder is limited by where the “corners” of the cylinder bump into the sides of the sphere. The problem asks you to find the largest volume such a cylinder inside a sphere can have if the sphere’s radius is 1.

Cylinder inside sphere has distance 1 to corner

Definite Integrals

Definition

Find the integral from 0 to 2 of x3 using the limit definition of the definite integral.

Reading ideas:

The limit definition says that a definite integral is the limit as the number of intervals increases indefinitely of the sum of interval widths times function values in those intervals.

Find the Integral:

Limit as n approaches infinity of sum of (2j/n)^3 (2/n) = 4

Clean Up Notation. x values are x0, x1, ..., xn; intervals are numbered 1, 2, ..., n. Sums are over areas, and so indexed from 1 to n.

Intervals numbered 1 to n with edges x_0 through x_n and sum from i = 1 to n

Example. Evaluate the integral from 2 to 3 of x2.

Integral from 2 to 3 of x^2 solved as sum by factoring out constants and using sum of powers rules

Take-Aways

You can evaluate definite integrals from the limit definition.

In particular

  1. Identify Δx, xi*, and f
  2. Write down specific sum (with limit)
  3. Find a closed form for the sum, using sum rules
  4. Take the limit, often using rules about limits of reciprocals of powers of n.

Problem Set

See handout for details.

Next

The Fundamental Theorem of Calculus, Part 1

Read section 5.3 from the beginning through the “Fundamental Theorem of Calculus Part 1: Integrals and Antiderivatives” subsection.

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