liggett_quizsolutions1,2

# liggett_quizsolutions1,2 - y = x and y = √ x about the...

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

Mathematics 31B/4 – Fall 2004 – Quiz Solutions (Note: The sketches that were required are not given below.) Quiz for Thursday, October 7 1. Use substitution to evaluate Z 13 0 dx 3 p (1 + 2 x ) 2 . Solution: Let u = 1 + 2 x,du = 2 dx . Then Z 13 0 dx 3 p (1 + 2 x ) 2 = 1 2 Z 27 1 u - 2 3 du = 3 2 u 1 3 27 1 = 3 . 2. Sketch the region between the curves y = 12 - x 2 and y = x 2 - 6, and ﬁnd the area of this region. Solution: The two parabolas meet at ( ± 3 , 3). Therefore, Area = Z 3 - 3 (18 - 2 x 2 ) dx = 18 x - 2 3 x 3 3 - 3 = 72 . Quiz for Thursday, October 12 1. Sketch and ﬁnd the area of the region bounded by the parabola y = x 2 , the tangent line to this parabola at (1 , 1), and the x -axis. Solution: The equation of the tangent line is y = 2 x - 1. Therefore, Area = Z 1 2 0 x 2 dx + Z 1 1 2 ( x - 1) 2 dx = 1 12 . Alternatively, one can integrate with respect to y and end up with a single integral: Area = Z 1 0 ± y + 1 2 - y dy = 1 12 . 2. Use slices to ﬁnd the volume of the solid obtained by rotating the region bounded by
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: y = x and y = √ x about the line y = 1. Be sure to sketch the region. Solution: Volume = π Z 1 • (1-x ) 2-(1-√ x ) 2 ‚ dx = π 6 . Quiz for Thursday, October 14 1. Use slices to ﬁnd the volume of the solid obtained by rotating the region bounded by y = x,y = 0 ,x = 2 , and x = 4 about the line x = 1. Be sure to sketch the region. Solution: Volume = π Z 2 (9-1) dy + π Z 4 2 ( 9-( y-1) 2 ) dy = 25 1 3 π. 2. Use shells to ﬁnd the volume of the solid obtained by rotating the region bounded by x = 4 y 2-y 3 and x = 0 about the x-axis. Be sure to sketch the region. Solution: Volume = 2 π Z 4 y (4 y 2-y 3 ) dy = 512 5 π....
View Full Document

## This note was uploaded on 04/29/2010 for the course MATH 31B 31B taught by Professor Liggett during the Spring '10 term at UCLA.

Ask a homework question - tutors are online