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PROBLEMS FOR EXAM IV--YOU WILL NEED TO DO THESE SAME
FOUR PROBLEMS ON THE EXAM WITH NO NOTES
| 1. Find the exact area of the region
bounded by the graphs of y = 0, y = 1 + x2/3, x = -1, and x = 3
by setting up the appropriate definite integral and evaluating it using
the Fundamental Theorem of Calculus. Approximate the area of the
same region by partitioning the interval [-1,3] into four equal length
sub-intervals and using four approximating rectangles with the height of
each approximating rectangle equal to the value of 1 + x2/3 at
the left endpoint of the sub-interval. (left
approximation) Also approximate the
area of the same region by partitioning the interval [-1,3] into four
equal length sub-intervals and using four approximating rectangles with
the height of each approximating rectangle equal to the value of 1 + x2/3
at the right endpoint of the sub-interval. (right
approximation) Look at the animation
below to see pictures for each of the two area approximations.
Compute [(left approximation) + (right
approximation)] / 2 to find an average
of these two approximations and compare this average to the exact area
computed using the Fundamental Theorem of Calculus. |
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| 2. Find the left and right sums
for the region bounded by the graphs of
f(x) = 1 + x2, y = 0, x
= -1, and x = 2.
Compute the limit as the number of subintervals
approaches infinity for each to find the area of the region. See
Example 4 on pages 264-5 in your textbook. You will only need to
do the "right" approximation on the exam.
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| 3. On a particular day when the
temperature varies greatly, the temperature in degrees Fahrenheit is given
by
where t is time in hours, with t = 0 representing
midnight. The hourly cost of cooling a house is $0.15 per
degree. Find the cost for the whole 24 hour day, starting at
midnight, of cooling the house if the thermostat is set at 72oF.
This idealized problem last appeared in the ninth edition of your textbook
as problem 123 on page 309. It was dropped from the 10th edition
perhaps because the scenario is unrealistic. The temperature inside
the house would probably remain below 720F for a period of time
after the temperature rose to 720F outside. |
T
 t
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4. Describe in words how the Trapezoidal Rule and
how Simpson's Rule work in approximating definite integrals. Include a
geometric description of how each rule works in approximating a definite
integral when the definite integral does represent the area under a curve over a
specified interval.
| The pictures at the right show the
"pieces of parabolas" used in applying Simpson"s Rule to
the function
f(x) = x3 - 2x2 - 3x + 10
over the interval [-2,2] with n = 2 (one
parabola) and n = 4 (two parabolas). In each of these cases
Simpson's Rule would yield the exact value of the integral.
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| The picture at the right illustrates
the Trapezoidal Rule with n = 4 (4 trapezoids). The Trapezoidal Rule
approximation to the integral given above with n = 4 would be 28 (compared
to 29.333...). |
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