lecture32hout

lecture32hout - MATH 006 Calculus and Linear Algebra...

Info iconThis preview shows pages 1–8. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: MATH 006 Calculus and Linear Algebra (Lecture 32) Albert Ku HKUST Mathematics Department Albert Ku (HKUST) MATH 006 1 / 18 Outline 1 Area under a graph 2 Definite Integrals 3 Definite integrals and Substitution 4 Average Value of a Continuous Function Albert Ku (HKUST) MATH 006 2 / 18 Area under a graph Area under a graph Question Given the graph of y = f ( x ), as shown in the figure below. How can we find the area S under the graph for a ≤ x ≤ b ? Albert Ku (HKUST) MATH 006 3 / 18 Area under a graph Riemann Sum Idea: Approximate S by rectangles! The sum of area of all such rectangles is called the Riemann sum . However, Riemann sum is just an approximation. How can we find the exact value of S ? Albert Ku (HKUST) MATH 006 4 / 18 Area under a graph “Limit” of Riemann Sum Another idea: Narrower rectangles give better approximation! Suppose y = f ( x ) = x 2 . We want to find the area under this graph for ≤ x ≤ 1. View this Java applet for details. The “limit” of Riemann sum is the area S ! Albert Ku (HKUST) MATH 006 5 / 18 Area under a graph Signed Area What is the limit of Riemann sum if the graph is not lying entire above the x-axis? The green area below the x-axis will be counted as “negative area” in the limit of Riemann sum. Hence, we have The limit of Riemann sum = signed area = Yellow area - Green area Albert Ku (HKUST) MATH 006 6 / 18 Definite Integrals Definite Integrals Definition The definite integral of f ( x ) from a to...
View Full Document

This note was uploaded on 12/28/2010 for the course MATH MATH006 taught by Professor Forgot during the Fall '09 term at HKUST.

Page1 / 18

lecture32hout - MATH 006 Calculus and Linear Algebra...

This preview shows document pages 1 - 8. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online