29-definite-integrals

29-definite-integrals - Math 1a The Definite Integral Fall,...

Info iconThis preview shows pages 1–4. 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
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Math 1a The Definite Integral Fall, 2009 We define the definite integral of y = f ( x ) from x = a to x = b as Z b a f ( x ) dx = lim n n X i =1 f ( x * i ) x where x i- 1 x * i x i . Note that x * i could be x i (in which case we have the limit of R n ) or x i- 1 (in which case we have the limit of L n ). One key point is that it doesnt matter which x * i we pick! Another is that, if f ( x ) 0, then the definite integral is an area. Compute the following definite integrals, using whatever x * i you like: 1 Z b x 2 dx 2 Z 1 x 3 dx Hint: n X i =1 i = n ( n + 1) 2 n X i =1 i 2 = n ( n + 1)(2 n + 1) 6 n X i =1 i 3 = n ( n + 1) 2 2 It can be simpler to compute a definite integral by recognizing that it represents an area. Use this to find the following definite integrals: 3 Z 10- 10 | x | dx 4 Z r- r r 2- x 2 dx 5 Z 10 x dx 6 Z 10- 10 x dx It is also useful to know the properties of the definite integral, as it can help to reduce integrals to known quantities. Here are eight properties of the definite integral: 1 Z b a c dx = c ( b- a ) 2 Z b a ( f ( x ) + g ( x )) dx = Z b a f ( x ) dx + Z b a g ( x ) dx 3 Z b a cf ( x ) dx = c Z b a f ( x ) dx 4 Z b a ( f ( x )- g ( x )) dx = Z b a f ( x ) dx- Z b a g ( x ) dx 5 Z c a f ( x ) dx + Z b c f ( x ) dx = Z b a f ( x ) dx 6 If f ( x ) 0 on [ a,b ], then Z a a f ( x ) dx 7 If f ( x ) g ( x ) on [ a,b ], then Z b a f ( x ) dx Z b a g ( x ) dx 8 If m f ( x ) M on [ a,b ], then m ( b- a ) Z b a f ( x ) dx M ( b- a ) Use these properties and the results from the first page to compute the following definite integrals: 7 Z 2 ( 4- x 2 ) dx 8 Z 2- 2 x + 4- x 2 dx 9 Z 5 ( 3 x 3- 7 x ) dx Now suppose that Z 5 f ( x ) dx = 3 , Z 10 5 f ( x ) dx =- 2 , Z 10 g ( x ) dx = 6 , and Z 5 g ( x ) dx =- 1 . Use the properties above to find the following integrals: 10 Z 5 10 f ( x ) dx 11 Z 10 f ( x ) dx 12 Z 10 (3 f ( x )- 2 g ( x )) dx 13 Z 5 ( g ( x )- 3 f ( x )) dx The Definite Integral Answers / Solutions 1 We find that Z b x 2 dx = 1 3 b 3 . Well use the right endpoint and compute R n , so x * i = x i = a + i x . In this case, x = b n and a = 0, so x i = i b n . Our function is f ( x ) = x 2 , so the right endpoint sum is R n = n X i =1 f ( x * i ) x = n X i =1 i b n 2 b n = b 3 n 3 n X i =1 i 2 . Using the hint, we know that n i =1 i 2 = n ( n +1)(2 n +1) 6 , so R n = b 3 n 3 n ( n +1)(2 n +1) 6 = b 3 6 n ( n +1)(2 n +1) n 3 . Thus Z b x 2 dx = lim n R n = lim n b 3 6 n ( n + 1)(2 n + 1) n 3 = b 3 6 2 1 = 1 3 b 3 , as claimed. 2 We find that Z 1 x 3 dx = 1 4 . This is very similar to Problem 1, with b = 1 and f ( x ) = x 3 . We get R n = n X i =1 i 1 n 3 1 n = 1 n 3 n X i =1 i 3 ....
View Full Document

Page1 / 6

29-definite-integrals - Math 1a The Definite Integral Fall,...

This preview shows document pages 1 - 4. Sign up to view the full document.

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