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    Integrals, Differential Equations and Limits

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    Question 1
    ∫ x3+4
    ________________________________________x2 dx
    Question 2
    Solve the initial value problem:
    ________________________________________dx = x^/¯(9+x2) ; y(-4) = 0
    Question 3
    Figure 3.1
    f(x) = x2+3

    Figure 3.2
    ni=1 f(xi) x

    Given Figure 3.1, find the exact area A of the region under f(x) on the
    interval [1, 3] by first computing Figure 3.2 and then taking the limit as

    Question 4
    Figure 4.1
    limn sin 
    ________________________________________4 + sin 2
    ________________________________________n + sin 3
    ________________________________________n + sin n

    Evaluate Figure 4.1 by interpreting it as the
    limit of Riemann sums for a continuous function f defined on [0, 1].
    Question 5
    Figure 5.1

    Given Figure 5.1, first sketch the graph y = f(x) on the given interval.
    Then find the integral of f using your knowledge of area formulas for
    rectangles, triangles and circles.
    Question 6
    Suppose that a tank initially contains 2000 gal of water and the rate of
    change of its volume after the tank drains for t min is V'(t) = (0.5)t - 30
    (in gallons per minute). How much water does the tank contain after it
    has been draining for 25 minutes?
    Question 7
    3∫1 6
    ________________________________________x2 dx
    Question 8
    Figure 8.1
    1∫0 1
    ________________________________________x+2 dx

    Find an upper and lower bound for the integral (Figure 8.1) using the
    comparison properties of integrals.
    Question 9
    Apply the Fundamental Theorem of Calculus to find the derivative of:
    h(x)= x∫2^/¯u-1dx
    Question 10
    4∫1 (4+^/¯x)2
    ________________________________________2^/¯x dx
    Question 11
    ∫2cos2 xdx
    Question 12
    Figure 12.1
    y = 9-x2 , y=5-3x

    Sketch the region bounded by the graphs of Figure 12.1, and
    then find its area.
    Question 13
    Figure 13.1

    Approximate the integral (Figure 13.1); n=6, by:
    a) first applying Simpsonfs Rule and
    b) then applying the trapezoidal rule.
    Question 14
    Find the mass M (in grams) of a rod coinciding with the interval [0, 4]
    which has the density function
    (x)= 5 sin 
    ________________________________________4 x
    Question 15
    The region R is bounded by the graphs
    x-2y=3 and x=y2
    Set up(but do not evaluate) the integral that gives the volume of the solid
    obtained by rotating R around the line x = -1.
    Question 16
    Figure 16.1

    Find the volume of the solid that is generated by rotating the region
    formed by the graphs of Figure 16.1 and y = 4x about the line x = 3.
    Question 17
    Use the method of cylindrical shells to find the volume of the solid
    rotated about the line x = -1 given the conditions:
    y = x3 - x2; y = 0; x = 0
    Question 18
    Find the length of the graph of
    y = 1
    ________________________________________3 x3/2 - x1/2
    (1, - 2
    ________________________________________3 )
    (4, 2
    ________________________________________3 )
    Question 19
    A 10-ft trough filled with water has a semicircular cross section of
    diameter 4 ft. How much work is done in pumping all the water over
    the edge of the trough? Assume that water weighs
    Question 20
    Figure 20.1
    y = x2

    The region in the first quadrant bounded by the graphs of y = x and Figure 20.1 is rotated around the line y = x. Find
    (a) the centroid of the region and
    (b) the volume of the solid of revolution.

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    Solution Summary

    A variety of calculus problems are solved. The solution is detailed and well presented.