ws29 (1)

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

This preview shows page 1. Sign up to view the full content.

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.)
This is the end of the preview. Sign up to access the rest of the document.

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 ....
View Full Document

## 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.

Ask a homework question - tutors are online