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# Integrals, Differential Equations and Limits

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Question 1
Find
&#8747; x3+4
________________________________________x2 dx
Question 2
Solve the initial value problem:
dy
________________________________________dx = x^/¯(9+x2) ; y(-4) = 0
Question 3
Figure 3.1
f(x) = x2+3

Figure 3.2
n&#61523;i=1 f(xi) &#61508;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
n&#8594;&#8734;

Question 4
Figure 4.1
limn&#61614;&#61605; sin &#61552;
________________________________________4 + sin 2&#61552;
________________________________________n + sin 3&#61552;
________________________________________n + sin n&#61552;
________________________________________n
________________________________________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
4&#8747;-2(|3-|3x||)dx

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
Evaluate
3&#8747;1 6
________________________________________x2 dx
Question 8
Figure 8.1
1&#8747;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&#8747;2^/¯u-1dx
Question 10
Evaluate:
4&#8747;1 (4+^/¯x)2
________________________________________2^/¯x dx
Question 11
Evaluate:
&#8747;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
1?0x4dx

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
&#61554;(x)= 5 sin &#61552;
________________________________________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
y=2x2

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
from
(1, - 2
________________________________________3 )
to
(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
62.5lb/ft3
Question 20
Figure 20.1
y = x2
________________________________________2

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.