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MATH 251, Section 516 : REVIEW
2nd Midterm Exam : November, 18th, 2014, 2 :20-3 :35 p.m
Policy for exam :
Write up your result, detail your calculations if necessary and BOX your final answer.
Your final work have to be neat, so use pencil first if you want.
Electronic devices are prohibited (cellphones, calculators, laptop or computer,. . .).
Advice : Read the whole exam before starting to answer, beginning by the questions which seem easiest
to you. Make sure you read the questions carefully.
No paper is necessary, you will be able compose on the same paper. Do not hesitate to
ask me scratch paper. Your copy shouldn’t be a draft !
The following is a summarize of the course, specified exercises are not exhaustive, those are just an idea
of the kind of practice which you should be able to do.
CHAPTER 13. : Sections 13.7 an 13.11 won’t be in the program !
1. The theory of integration via Riemann sum is to know but its practice isn’t usually used.
2. Exercices : 26, 27, 31.
– How to compute a double integral over a rectangle.
– Be comfortable with the antiderivatives (Know the classical ones and trigonometric formulas as
linearization of cosine and sine).
– Use Fubini’s Theorem.
– Find the volume of a solid bounded by 2 surfaces or by a surface and planes.
3. Exercices : 13,19, 20, 22, 39, 42.
– Sketch the bounded region of integration.
– How to integrate a function of 2 variables over a general bounded region of R2 .
– Know how to switch the order of integration in some cases.
– Find a volume between 2 surfaces or planes and a surface.
– Know the properties of multiple integrals (linearity ,. . .).
4. Polar coordinates. Find cartesian coordinates from polar coordinates and conversely.
5. Exercices : 2,3, 9, 15, 20.
– Know when to use polar coordinates to describe a bounded region of R2 (depends on the domain
of integration and the function to integrate (of the form distance from the origin).
– Use the change’s theorem in polar coordinates in a double integral (don’t forget the multiplication
by r).
6. Exercices : 5,7,8. Applications of double integrals (formulas to determine the mass of a object in R2
with a non constant density, the coordinates of the center of mass, the moments of inertia).
8. Exercices : 17,20, 25,38. triple integrals and its applications (know the same things as in Section 13.2,
13.3, 13. 6 for a triple integral).
– Use Fubini’s Theorem for triple integrals.
– Determine and Sketch the domain of integration (if this one is not too complicated).
– Integrate over a general bounded region of the space, switch the order of integration.
– Applications of triple integrals : compute volume of a bounded region of the space, determine the
mass of a solid object, determine the center of mass of a solid object with a non constant density
and formulas for the inertia moments.
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9. Know the formulas to transform cartesian coordinates to cylindrical coordinates and conversely and
to obtain the spherical coordinates.
10. Exercices : 1,3,4,5,6,9, 11,15.
– Know when use cylindrical or spherical coordinates (to describe the domain and depend on the
function to integrate). Use them to describe the appopriate bounded region in the space.
– Know very well the change’s theorem in cylindrical coordinates or in spherical coordinates (don’t
forget the multiplication in the integral of r or respectively for spherical coordinates ρ2 sin φ.)
– Applications : Know to sketch the integration region from a triple integral under these changes of
coordinates, compute a volume under theses changes, . . .
A review p. 862 of this chapter in the textbook should be seen. . .
CHAPTER 14. :
1. Exercices : 5,7,10,11. What is a vector field ? How to sketch it ?
2. Exercices :4,6, 16, 35, 36.
– How to parametrize a curve in R2 or in R3
– Use those parametric equations to compute a line integral with respect to the arc length in R2 or
the space R3 .
– Compute line integrals with respect to the variable x, or y or z.
– Compute the work of a force along a curve ( given by a line integral of a vector field). Be careful
in this case, the orientation of the curve counts.
3. Exercices : 1,4, 9, 15,19.
– What is a conservative vector field ? What can we say on the work of a conservative force for
example ? know the fundamental theorem for line integrals and its interpretation.
– What does to be independent of path mean ?
– Know the different types of curves and domains.
– How to determine when a vector files in the plane is conservative or not ?
4. Exercices : 2,3, 16, 21. Know the Green’s theorem and use it.
5. Exercices : 14,17, 18. Definition of the Curl and the divergence and its applications to determine if
a vector field in the space is or not conservative and if we can determine a potential function for the
curl. Know the second formulation of the Green’s theorem with the operator curl.
6. Exercices : 1,3, 6, 15, 17. Parametric surfaces and their areas.
– Find a parametric equation for a surface (ellipsoid, cylinder, paraboiloid.
– Thanks to this parametrization, find the tangent plane to the surface at a point.
– Find the surface area.
A review section is also available at the end of Chapter 14, take a look to practice more.