ws29 (1) - a Z x √ x 2 5 dx b Z 1 √ x 2 x 4 3 2 dx c Z...

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WORKSHEET 29 - Fall 1995 1. Label each of the following as “TRUE” or “FALSE:” a) Z ( x 2 +1) 9 (2 x ) dx = ( x 2 +1) 10 10 + C c) Z sin( x )cos( x ) dx = sin 2 ( x ) 2 + C e) Z 2tan( x )sec 2 ( x ) dx =tan 2 ( x )+ C b) Z 2 x dx = 2 x +1 ( x +1) + C d) Z sin( x )cos( x ) dx = cos 2 ( x ) 2 + C f) Z 2tan( x )sec 2 ( x ) dx =sec 2 ( x )+ C 2. Evaluate the following de±nite integrals: a) Z 3 1 ± t 2 t ²± t + 2 t ² dx d) Z 1 4 0 tan 2 ( πx ) dx g) Z 5 0 p y +1 dy b) Z 1 0 x 3 x 4 +9 dx e) Z π 2 0 (cos(2 x )) 3 sin(2 x ) dx h) Z 3 1 17 x 2 +17 x - 2 dx c) Z 2 0 1 ( x +1) 2 dx f) Z 2 π 0 p 1 cos 2 θdθ i) Z 4 1 p 2+ x x dx 3. a) The region under the graph of y = 2 x +4on[ 2 , 1] is to be divided into two parts of equal area by a vertical line. Where shopuld the line be drawn? b) Where would you draw a horizontal line to divide the region in part a) into two parts of equal area? 4. Consider the area enclosed by the curve y = x 2 ,the x -axis, the lines x =1and x = 2. What vertical line will divide this area into two equal parts? 5. Evaluate three of the following four integrals (one cannot be done using methods we know so far.)
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Unformatted text preview: a) Z x √ x 2 + 5 dx b) Z 1 ( √ x 2 + x + 4) 3 / 2 dx c) Z 2 x + 1 ( x 2 + x + 3) 3 dx d) Z ± x − 3 x ² 5 ± 1 + 3 x 2 ² dx 6. a) Prove that if f is integrable and nonnegative on the closed interval [ a, b ] then R b a f ( x ) dx ≥ 0. b) Use part a) to prove that if f and g are integrable on the closed interval [ a, b ] and f ( x ) ≤ g ( x ) ∀ x ∈ [ a, b ] then Z b a f ( x ) dx ≤ Z b a g ( x ) dx. c) Show that − 3 < Z 2-1 x 2 − 1 x 2 + 1 dx < 2 . (Hint: Use a) or b), ±nd minimum and maximum values of the integrand on [ − 1 , 2].) d) Show that 1 7 √ 2 ≤ Z 1 x 6 √ 1 + x 2 dx ≤ 1 7 ....
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This note was uploaded on 04/11/2011 for the course MATH 1400 taught by Professor Grether during the Spring '08 term at North Texas.

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