MATH 126 midterm 2 preparation guide
Midterm 2 will cover sections 6.6, 7.1, 7.2, 7.3, 7.4, 7.5, 7.6, 8.1. Important concepts and methods include:
ˆ Two types of improper integrals: type 1 when the integration interval is infinite and type 2 when a
function is not continous in some points of the interval (6.6)
ˆ Area between curves. Expressing the area as an of distance between curves (7.1)
ˆ Expressing the volume as integral of the cross-section area function (7.2)
ˆ Method of cylindrical shells to compute the volume of rotational figures (7.3)
ˆ Arc length formula to compute the length of the arc of the graph of the function(7.4)
ˆ Area of surface of revolution. Expressing the area as the integral of the differential of the arc length
function (formulas 7, 8 of 7.5). Particular cases of the formula: when a curve is given as y = f (x),
x = g(y), and the rotation goes about x-axis and y-axis (7.5)
ˆ Work. Expression as integral of the force (formula 4 of 7.6). COnstructing the integrals expressing
work (examples 3,4 of 7.6)
ˆ Hydrostatic force and pressure. Expressing hydrostatic force as an integral of the pressure (examples
5,6 of 7.6)
ˆ Convergent sequences and limits. Relation between limit of sequence and limit of functions (Theorem
3 of 8.1)
ˆ Squeeze theorem for sequences, Continuity and convergence theorem, monotonic sequence theorem
(8.1)
The list of typical exercises:
ˆ Section 6.6, Improper integrals: convergent and divergent integrals.
Typical exercise is to determine if the given integral is convergent or divergent, for example:
Z ∞
Z 1
Z 1
Z ∞
3 + e−x
1
x4
√
dx,
dx,
dx,
x2 ln(x)dx
1
6 − 2x + 1
x
3
x
(x
−
1)
0
0
0
0
ˆ Section 7.1, Areas between curves
Typical exercises: Find the region bounded by curves:
y 2 = 4 − x and x + y = 2
x = y4 , y =
√
2 − x, and y = 0
ˆ Section 7.2, 7.3, Volumes.
Typical exercises:
The region bounded by curves x − y = 1, y = x2 − 4x + 3 is rotated about the axis y = 3. Find
the volume of the resulting solid.
√
The region bounded by curves y = x, y = 0, x = 1 is rotated about the axis x = −1. Find the
volume of the resulting solid.
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– Section 7.4, Arc length Typical exercise: The curve is given by equation y = x2 −
1 6 x 6 2. Find the exact length of the curve.
1
2
ln(x) where
– Section 7.5, Area of surface of revolution Typical exercises:
The curve y = (x − 1)1/3 , 1 6 y 6 2 is rotated about the axis x = −1. Find the area of
surface of revolution.
Consider a circle of radius 1 and center (2, 0). A torus is obtained by rotating the given circle
about the y-axis. Find the surface area of the torus.
– Section 7.6, Work and hydrostatic pressure
Typical exercises:
A 10-ft chain weighs 25 lb and hangs from a ceiling. Find the work needed to lift the lower
end of the chain so that its level with the upper end.
A cylindrical tank with diameter 3 m is filled with water. Find the force exerted by water on
one end of the tank if:
(a) The tank is full
(b) The tank is one-half full
– Section 8.1, Sequences
Typical exercise: Determine if the given sequence is convergent. If it is, find its limit, for example:
an = (1 + n12 )n , an = ln(2n + 1) − ln(n + 1/n), an = 2−n sin(n2 ).
Another good source of practice problems can be found in the list of old final exams:
https://dornsife.usc.edu/mathcenter/126/
External tutorials:
http://tutorial.math.lamar.edu/Problems/CalcII/CalcII.aspx
http://tutorial.math.lamar.edu/ProblemsNS/CalcII/CalcII.aspx
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