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Math 156 Review topics for Exam 2
Improper integrals:
What makes an improper integral improper. The definition of the value of an improper integral in terms of limits, evaluating improper integrals this way.
Using L’Hopital’s rule to evaluate resulting limits 0
xexdx , 0
1 ln x dx Special improper integrals you should know (and be able to show):
xp^ dx^ converges for^ p^ ^ 1,^ diverges otherwise, for example
x dx^ ,^ 0
x
dx , 0
x^2
dx
xp^ dx^ converges for^ p^ ^ 1 ,^ diverges otherwise, for example
x dx^ ,^ 1
x
dx , 1
x^2
dx
ex^ dx 1 converges , 0
1
ln x dx 1 converges , 0
xex^ dx Comparison (for positive integrands) Roughly speaking: if an integral converges, any smaller integral converges if an integral diverges, any larger integral diverges
To analyze, determine the dominant term in a sum when you approach the discontinuity, or as you go to . That will usually tell you whether or not the integral converges.
x ex^ dx^ ,^ 0
(^1 1) ex x x
dx , 1
x x^2
dx , 0
(^1) ln x 1 x
dx , 0
ex
2 dx
Area between curves: Pay careful attention to specification of the region in question. You may need to find points of intersection yourself. Depending on shape of the region, it may be more convenient to use vertical strips and integrate with respect to x, or to use horizontal strips, integrating with respect to y.
Volumes:
By method of slices/cross sections: V Axdx
Typical problem: We specify the base and describe cross-sections perpendicular to one of the axes. You figure out the dimensions of a general cross section and then the area. Obviously, you might find yourself integrating with respect to y as well as with respect to x.
Volumes of revolution: Determine the region to be rotated. The shape of the region (i.e. the bounding curves) and the functional form of those curves (is y in terms of x, can we solve for x in terms of y? will usually suggest whether you want vertical strips (integration with respect to x) or horizontal strips (integration with respect to y). Then the axis of rotation will determine which of the following approaches applies:
Disks/washers: When you rotate strips perpendicular to the axis of rotation you get washers (or disks, if one end of the strip is on the axis of rotation)
V rout^2 rin^2 dx (rotation axis of form y c, such as x axis, y 0 )
The integrand represents the area of a washer (a disk with a hole removed) times the thickness (dx)
Cylindrical shells: When you rotate strips parallel to the axis of rotation you get cylindrical shells
V 2 rdx (rotation axis of form x c, such as y axis, x 0 )
Above, r is the distance of the strip from the axis, represents the length of the strip ( ytop ybot when rotating about axis x c. ) Integrand represents area dx of rectangular strip times circumference 2 r of circle when strip is rotated.
In applying these formulas to specific problems we often make use of notation such as xright , xleft, ytop , ybot
Ex: Rotate region in first quadrant bounded by y x^3 x , y 0, x 1 about the a) x axis: Here integrating with respect to x and using washers is very easy. b) y axis: Integrating with respect to y and using washers is hard to do, because we can’t solve for x in terms of y. Instead, integrate with respect to x and use cylindrical shells.
In this example the functional form of the curves is driving the integration method.
Region in first quadrant bounded by y x^2 , y 2 x^2 4, y 0. This was on the quiz. Here the geometry of the region suggests that using horizontal strips is easiest and you can do it because you can solve for x in terms of y easily.
Region bounded by y x^2 , y 2 x^2 4, x 0 Here the geometry is more suited for vertical strips.
Note that we sometimes want to rotate about a line like x 2, y 3, etc and then you have to carefully identify the various lengths and radii in the formulas.
Arc length: s x
1
x 2 1 dydx
2 dx Sometimes we write ds dx^2 dy^2 and then this
can be converted into either the integral at the left, when integrating with respect to x, or the
integral s y
1
y 2 1 dxdy
2 dy if integrating with respect to y.
Work: A force Fx that depends on position x, exerted over some interval a x b,
produces work W a
b Fxdx. Examples include stretching a spring, compressing a piston.
Work problems also appear, e.g. when building a wall, pulling a hanging chain up, pumping out a liquid. In these cases the material is to be redistributed in space and each little bit of material requires a little bit of work to move it where it is supposed to go.
Pressure and total force: P gh is the pressure at depth h of a liquid of density , where g is the acceleration of gravity in the appropriate units. Force is pressure times area. We sometimes write w g as the "weight density", the number of pounds per unit volume
