w7-C - Math 20C Multivariable Calculus Lecture 19 1 ' $...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
Math 20C Multivariable Calculus Lecture 19 1 Slide 1 $ % Double integrals (Sec. 15.1 - 15.2) Review of the integral of single variable functions. DeFnition of double integral on rectangles. Average of a function. Double integrals general domains (Sec. 15.2). Examples of double integrals. Slide 2 $ % Integral of a single variable function Defnition 1 (Integral oF single variable Functions) Let f ( x ) be a function deFned on a interval x [ a, b ] . The integral of f ( x ) in [ a, b ] is the number given by Z b a f ( x ) dx = lim n →∞ n X i =0 f ( x * i ) Δ x, if the limit exists. Given a natural number n we have introduced a partition on [ a, b ] given by Δ x = ( b - a ) /n . We denoted x * i = ( x i + x i - 1 ) / 2 , where x i = a + i Δ x , i = 0 , 1 , ··· , n . This choice of the sample point x * i is called midpoint rule.
Background image of page 1

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

View Full DocumentRight Arrow Icon
Math 20C Multivariable Calculus Lecture 19 2 Slide 3 $ % Double integrals on rectangles Defnition 2 (Double integrals on rectangles) Let f ( x, y ) be a function deFned on a rectangle R = [ x 0 , x 1 ] × [ y 0 , y 1 ] . The integral of f ( x, y ) in R is the number given by Z Z R f ( x ) dxdy = lim n →∞ n X i =0 n X j =0 f ( x * i , y * j ) Δ x Δ y, if the limit exists. Given a natural number n we have introduced a partition on R by rectangles of side
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 12/26/2009 for the course MATH 20C taught by Professor Helton during the Fall '08 term at UCSD.

Page1 / 8

w7-C - Math 20C Multivariable Calculus Lecture 19 1 ' $...

This preview shows document pages 1 - 3. Sign up to view the full document.

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