Group 1
Group 4
Group 7
Group 2
Group 5
Group 8
Group 3
Group 6
Group 9
MATH 126 - 63991 Lab Section
Week 2: 1/24 and 1/26
Group 10
Group 11
Group 12
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Example (1) Why is u = cos(x) and Zdv = x dx a poor choice for
Integration By Parts when evaluating x cos(x) dx?
Example (2) For each of the following expressions, determine whether
Substitution or Integration By Parts is the most appropriate method to
evaluate the integral. Briefly explain your reasoning.
Z
Z
2
(A1)
x cos(x ) dx
(A2)
x cos(x) dx
Z
(B1)
2 x
Z
x e dx
(B2)
2
x e x dx
Example (3) Abraham usesZthe substitution u = tan(x) and Savannah
uses v = sec(x) to evaluate tan(x) sec2 (x) dx. Show that they obtain
different answers and explain why this is not a contradiction.
MATH 126 - 63991 Lab Section
Week 2: 1/24 and 1/26
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Example (4) Evaluate the following indefinite integrals:
Z
Z
2 x
x 3 dx
x arctan(x) dx
(i)
(v)
Z
(ii)
Z
(iii)
x 3 sin(3 − x) dx
x
e sin(2x) dx
Z
ln |x| dx
MATH 126 - 63991 Lab Section
Z
(ln |y |)2 dy
Z
√
x 2 x − 1 dx
(vii)
Z
(iv)
cos(x) ln | sin(x)| dx
(vi)
(viii)
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Example (5) Two cars, A and B, start side-by-side and
accelerate from rest. The graphs of their velocity
functions are given below.
(i) Which car is ahead at time a? Explain.
(ii) Interpret the area of the shaded region in physical
terms.
(iii) At what times are the cars side-by-side?
MATH 126 - 63991 Lab Section
Week 2: 1/24 and 1/26
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Example (6) Sketch the region enclosed by the given
curves. Decide whether to integrate with respect to x or
y. Draw a typical approximating rectangle and label its
height and width. Then find the area.
(i) y = 2x + 3, y = 13 − x 2 , x = −1, and x = 2.
(ii) x = 45 − 5y 2 and x = 5y 2 − 45.
(iii) y = 6 cos(πx) and y = 12x 2 − 3.
(iv) y =
√
x, y = 12 x, and x = 25.
(v) y = |3x| and y = x 2 − 4.
MATH 126 - 63991 Lab Section
Week 2: 1/24 and 1/26
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Example (7) Find the number b such that the line
y = b divides the region bounded by the curves
y = 4x 2
and
y = 16
into two regions with equal area.
Example (8) Sketch the region under the graph of
f (x) =
1
1 + x2
and then show that the total area contained between the
curve and the x-axis is π units2 .
MATH 126 - 63991 Lab Section
Week 2: 1/24 and 1/26
The University of Kansas
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