PP 8.7 - Approximate Integration We want to develop more...

Info iconThis preview shows pages 1–7. Sign up to view the full content.

View Full Document Right Arrow Icon
Approximate Integration
Background image of page 1

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

View Full DocumentRight Arrow Icon
We want to develop more methods to approximate the value of a definite integral when it cannot be found directly using the Fundamental Theorem of Calculus. Methods known: Left-endpoint approximation Right-endpoint approximation = - = n i i n x x f L 1 1 ) ( = = n i i n x x f R 1 ) ( Midpoint approximation ) ( , ) ( 1 2 1 1 i i i n i i n x x x x x f M + = = - = n a b x x x f dx x f n i i b a - = = , ) ( ) ( : Know 1 *
Background image of page 2
Another approximation, called the Trapezoidal Rule, results when averaging the right and left endpoints approximations: [ ] [ ] ) ( ) ( 2 ... ) ( 2 ) ( 2 ) ( 2 ) ( ) ( 1 2 1 0 1 1 1 2 1 2 1 n n n i i n i i n n n x f x f x f x f x f Δx x x f x x f L R T + + + + + = + = + = - = = - Area of a trapezoid = ) ( 2 1 2 1 b b h + Why Trapezoidal Rule?
Background image of page 3

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

View Full DocumentRight Arrow Icon
) ( ) ( 2 1 1 i i x f b x f b x h = = = - [ ] ) ( ) ( ) ( Area 1 2 1 2 1 2 1 i i x f x f x b b h + = + = -
Background image of page 4
Comparing the Methods Under estimate Over estimate
Background image of page 5

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

View Full DocumentRight Arrow Icon
Over estimate Better estimate?
Background image of page 6
Image of page 7
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 21

PP 8.7 - Approximate Integration We want to develop more...

This preview shows document pages 1 - 7. Sign up to view the full document.

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