MA 126: Calculus II
Midterm Test #2, March 21, 2005
Time limit: 105 min.
Your name (print):
Your signature:
1. How large do we have to choose nso that the approximations Tnand Mnto the
integral
Z√π
0
sin(x2)dx
are accurate within 0.0001?
10 points
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Calculus II Midterm Test 2, March 21, 2005: Solutions and Questions, Exams of Calculus
The calculus ii midterm test held on march 21, 2005, with 10 questions covering topics such as approximations, improper integrals, bisection method, volume of solids, length of curves, and convergence of sequences and series. Students are required to find the answers to each question within the given time limit.
Typology: Exams
2012/2013
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MA 126: Calculus II Midterm Test #2, March 21, 2005
Time limit: 105 min. Your name (print):
Your signature:
- How large do we have to choose n so that the approximations Tn and Mn to the integral
∫ √π
0
sin(x^2 )dx
are accurate within 0.0001?
- Calculate the improper integral
∫ (^) ∞
1
2 −xdx.
- Find the number a such that the line x = a bisects the area under the curve
y =
x
, 1 ≤ x ≤ 1 .44. Also, find the number b such that the line y = b bisects the
above area.
- Consider the region D bounded by the curves y = x^2 /^3 , x = 8, y = 0. Find the volume of the solid S that can be obtained by revolving D about the y-axis.
- Find the centroid of the lamina (with constant density) bounded by the curves y = 1/x, y = 0, x = 1, x = 4.
- Let an = cos
( (^) π
2 n
. Check the sequence {an} for monotonicity and boundedness.
Is it convergent? Explain everything!
- Find the values of x for which the series
∑^ ∞
n=
3 n(x + 2)n+
converges. Find the sum of the series for those values of x.