30-calculating-definite-integrals

30-calculating-definite-integrals - Math 1a Calculating...

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Unformatted text preview: Math 1a Calculating Definite Integrals Fall, 2009 & $ % The Definite Integral If F ( x ) is any anti-derivative of f ( x ) (that is, if F ( x ) = f ( x )), then the definite integral of f ( x ) from a to b is Z b a f ( x ) dx = F ( x ) b a = F ( b )- F ( a ) . Compute the following definite integrals: 1 Z 6 1 (4 x + 2) dx 2 Z 9 1 x dx 3 Z 1 ( x- x 2 ) dx 4 Z 3 ( x 2- 4 x ) dx 5 Z e 1 1 x dx 6 Z 5- 5 ( x 3 + x ) dx 7 Z 1 2 x dx 8 Z 1 e x dx 9 Z 1 a x dx , a > 10 Z / 4 sec( x ) tan( x ) dx 11 Z 1 1 1 + x 2 dx 12 Z 1 / 2 1 1- x 2 dx 13 Z / 4 1 + cos 2 ( x ) cos 2 ( x ) dx 14 Z- 3 e x +3 dx 15 Z 2 1 x 2 + 2 x- 3 x dx 16 (a) Verify that d dx ( x ln( x )- x ) = ln( x ). (b) Use part (a) to compute Z e 1 ln( x ) dx . 17 What is wrong with the following computation of the area under y = 1 /x 4 from x =- 1 to x = 1? Area = Z 1- 1 x- 4 dx = x- 3- 3 1- 1 = 1- 3-- 1- 3 =- 2 3 . (Notice that the function is always positive, yet the area underneath the curve is negative! So there must be something wrong.) 18 Find the area of the region bounded by the give curve and the x-axis over the given interval: (a) y = cos( x ),- 2 x 2 (b) y = x 3- 4, x 2 Calculating Definite Integrals Answers / Solutions 1 Z 6 1 (4 x + 2) dx = ( 2 x 2 + 2 x ) 6 1 = ( 2(6) 2 + 2(6) )- ( 2(1) 2 + 2(1) ) = 80 2 Z 9 1 x dx = x 3 / 2 3 / 2 9 1 = 2 3 ( 9 3 / 2- 1 3 / 2 ) = 52 3 17 . 3333 3 Z 1 ( x- x 2 ) dx = 1 2 x 2- 1 3 x 3 1 = 1 6 4 Z 3 ( x 2- 4 x ) dx = 1 3 x 3- 2 x 2 3 = 1 3 3 3- 2(3) 2- 1 3 3- 2(0) 2 =- 9 5 Z e 1 1 x dx = ln( x ) e 1 = ln( e )- ln(1) = 1 6 Z 5- 5 ( x 3 + x ) dx = 1 4 x 4 + 1 2 x 2 5- 5 = 1 4 (5) 4 + 1 2 (5) 2- 1 4 (- 5) 4 + 1 2 (- 5) 2 = 0 7 Z 1 2 x dx = 2 x 1 ln(2) 1 = 2 1- 2 ln(2) = 1 ln(2) 8 Z 1 e x dx = e x 1 = e 1- e = e- 1 9 Z 1 a x dx = a x 1 ln( a ) 1 = a 1- a ln( a ) = a- 1 ln( a ) unless a = 1, in which case this is the integral of y = 1 from...
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30-calculating-definite-integrals - Math 1a Calculating...

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