Midterm Exam 3 Checklist
The exam will cover all of Chapter 15 and sections 16.1-16.3.
The exam problem will be similar to homework problems.
Summary of concepts
Sections 15.1-15.3: Double integrals
- Double integral over a rectangular region.
- Double integral over a rectangular region
if the integrand has the form f(x)g(y).
- Double integrals over “vertically simple”
(Type I) and “horizontally simple”(Type II) regions.
- Reversing the order of integration.
- Area as a double integral.
- Volume as a double integral.
- Average value of a function.
Section 15.4: Double integrals in polar coordinates
- How to write the area element dA = dx dy in polar
coordinates.
- Computation of double integrals in polar coordinates.
- Converting double integrals from rectangular coordinates to
polar coordinates.
Section 15.5: Applications of double integrals
- Mass density. Mass of a flat lamina.
- Center of mass (centroid) and how to compute it.
- Moments of inertia and radii of gyration and how to compute them.
- Probability: joint density function.
- How to compute probability by double integrals.
- Expected values of random variables.
Section 15.6: Triple integrals
- Triple integrals in rectangular coordinates.
- Projecting a solid onto a coordinate plane.
- To project a curve of intersection of two surfaces
onto a coordinate plane - eliminate the third variable.
- Sketching tree dimensional regions and their
projections to coordinate planes.
- Iterated integrals with different orders of integration.
- Volume by triple integrals.
- Mass and mass density in three dimensions.
- Center of mass (centroid) in three dimensions
and how to compute it.
- Moments of inertia in three dimensions
and how to compute them.
Sections 15.7-15.8: Triple integrals in cylindrical
and spherical coordinates
- The geometry of cylindrical and spherical coordinates,
the formulas for converting back and forth.
- Volume element dV translated into cylindrical and spherical
coordinates.
- How to compute an integral by changing to cylindrical or
spherical coordinates.
- Converting a triple integral from cartesian to cylindrical or
spherical coordinates.
Section 15.9: Change of variables in multiple integrals
- Jacobian determinant associated to a transformation/change
of variables.
- Change of variables formulas in two and three dimensions.
- Given an integral in (x,y) coordinates, and a change of
variables to (u,v) coordinates, how to express a region in the (x,y)
plane in terms of a region in the (u,v) plane.
- How to compute the integral of a given function f(x,y) over a
given region D by a change of variables.
Section 16.1: Vector Fields
- Geometric and physical meaning of a vector field.
- Plots of vector fields in the plane.
- Gradient vector field of a function.
- Conservative vector fields and potential functions.
- How to find a potential function of a vector field.
Section 16.2: Line integrals
- Definition and evaluation of a line integral with respect to
arclength ds.
- Definition and evaluation of a line integral with respect to
coordinate variables dx, dy, dz.
- Independence of parametrization.
- Physical meaning of the line integral as the work of a force.
- How line integrals along the curves C and -C are related.
- Line integral along a piecewise smooth path.
Section 16.3: The fundamental theorem for line integrals
- Connected regions and simply connected regions (with no holes).
- Independence of path, definition.
- A line integral is independent of path if and only if
the integral over every closed path is zero.
- Independence of path is equivalent to the vector field
being conservative.
- How to tell whether a vector field is conservative.
- How to compute a potential of a conservative vector field
by means of line integral.
Typical computational tasks
- Iterated integrals (15.2-15.3).
- Evaluation of iterated integrals by reversing the order
of integration (Sketch the region!) (15.3).
- Areas and volumes by double integrals (15.3).
- Evaluating double integrals in polar coordinates (15.4).
- Mass, center of mass (centroid), and moments of inertia
of a lamina (15.5).
- Evaluating triple integrals (15.6).
- Volumes by triple integrals (15.6).
- Mass, center of mass, and moments of inertia of a solid (15.6).
- Conversion between rectangular, cylindrical, and spherical
coordinates (15.7-15.8).
- Evaluating triple integrals in cylindrical and spherical
coordinates (15.7-15.8).
- Change of variables in double integrals (15.9).
- Compute the gradient of a function (16.1).
- Evaluate a line integral (two types) (16.2).
- Use a line integral to compute mass, center of mass,
or moments of inertia of a wire (16.2).
- Determine whether a vector field is conservative (16.3).
- Find a potential function for a conservative vector field
(two methods) (16.3).
- Show that a given line integral is independent of path (16.3).
Things to remember
- Any definitions and formulas you need for the kinds of
calculations mentioned above. The best way to memorize formulas
is to learn how they are obtained. If you understand the formula,
you will memorize it easily.