Integral Converges - Calculus II - Exam, Exams of Calculus

Main points of this exam paper are: Integral Converges, Midpoint Approximation, Subdivisions of Interval, Value of Integral, Area Bounded by Graphs, Numerical Value, Volume of Solid, Integral Diverges, Arc Length of Graph

Typology: Exams

2012/2013

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MA 126-E0 Summer 2006 Test 1
1. Evaluate R1
0(3x+ 1)e๎˜€xdx.
2. Evaluate Z4
3
x๎˜€9
x2+ 3x๎˜€10dx:
3. Evaluate R1
0x2(1 + 2x3)5dx.
4. Suppose we want to approximate R5=2
2sin(x2)dx.
(a) Find a value of nso that the midpoint approximation using nsubdivisions
of the interval will be within 1=100 of the value of this integral.
(b) Write out the terms of the midpoint approximation for this value of n.
(Do not attempt to evaluate this sum).
5. Find the area bounded by the graphs of y= sin(x)and y= 1=2for
0๎˜”x๎˜”๎˜™=2.
6. Determine whether the following integral converges ot diverges:
Z1
1
1
(3x+ 1)2dx:
If the integral converges, determine its value. If the integral diverges, it has no
numerical value.
7. Let Dbe the region bounded by the graphs of y=e2x,x=๎˜€1and x= 2,
and the x๎˜€axis. Find the volume of the solid that results if Dis rotated about
the x๎˜€axis.
Each problem is worth 16 points. In Problem 4, (a) is worth 10 points, and
(b), 6points.
1

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MA 126-E0 Summer 2006 Test 1

  1. Evaluate

R 1

0 (3x^ + 1)e

xdx.

  1. Evaluate (^) Z (^4)

3

x 9 x^2 + 3x 10

dx:

  1. Evaluate

R 1

0 x

(^2) (1 + 2x (^3) ) (^5) dx.

  1. Suppose we want to approximate

R 5 = 2

2 sin(x

(^2) )dx. (a) Find a value of n so that the midpoint approximation using n subdivisions of the interval will be within 1 = 100 of the value of this integral. (b) Write out the terms of the midpoint approximation for this value of n. (Do not attempt to evaluate this sum).

  1. Find the area bounded by the graphs of y = sin(x) and y = 1= 2 for 0  x  = 2.
  2. Determine whether the following integral converges ot diverges: Z (^1)

1

(3x + 1)^2

dx:

If the integral converges, determine its value. If the integral diverges, it has no numerical value.

  1. Let D be the region bounded by the graphs of y = e^2 x, x = 1 and x = 2, and the x axis. Find the volume of the solid that results if D is rotated about the x axis. Each problem is worth 16 points. In Problem 4, (a) is worth 10 points, and (b), 6 points.