Test 3 Concepts

Test 3 Covers Sections - 12.2-12.7, 13.1-13.4, HW16 - HW21
(Old Book 15.3-15.5, 15.7-15.8, 16.1-16.4)

Section 12.1-12.2 Double Integrals (old book 15.1-15.3)
1) Know how to set up the x and y bounds for a region in the xy-plane.
2) Know that constant bounds must go on the outside integral.
3) Know that you must integrate multiple integrals from the inside integrals outwards and the dx, dy, dr, etc.. must match the integral limits from the inside outwards.
4) Know that x is bounded by a constant if it is bounded by a vertical line and y is bounded by a constant if it is bounded by a horizontal line.
5) dA = dx dy, or dA = dy dx
6) Know that Area of a region D is:
           A(D) = SS 1 dA
                       D
7) Know that you can multiply multiple integrals if the bounds are all constant and the functions can be separated by variables:
         SS f(x,y) dx dy = S g(x) dx * S h(y) dy

Section 12.3 Polar Regions (old book 15.4)
1) Know when to integrate using dA = r dr d(theta); i.e. when the region is a circle or part of a circle or the region is defined by a polar function.
2) Know how to find the bounds for r and theta.

Section 12.4 Word Problems (old book 15.5)
1) Know how to set up mass, moments about axes, moments of inertia, and center of mass using the given formulas.
2) Know what a lamina is and why density and mass are in terms of units-squared rather than units-cubed.

Section 12.5 Triple Integrals (old book 15.7) 
1) Know that dV = dx dy dz for rectangular coordinates, or any order of "dx dy dz."
2) Know how to set up the bounds for any rectangular triple integrals. 
3) Know that the Volume of a 3D object E is:
        V(E) = SSS 1 dV
                     E

 Section 12.6 Triple Integrals in Cylindrical Coordinates (old book 12.7/15.7) 
1)  Know that you use cylindrical coordinates when the projection of the 3D object E onto one of the coordinate planes is circular or a polar region.
2)  Know that dV = dz r dr d(theta) (or dV = dy r dr d(theta) or dV = dx r dr d(theta)).
3)  Know how to set up the bounds for any cylindrical triple integrals.


Section 12.7 Triple Integrals in Spherical Coordinates (old book 12.7/15.8)
1)  Know that you use spherical coordinates when the 3D object E is part of a sphere.
2)  Be able to describe what the variable "phi" and "p" are and be able to find their bounds.
3)  Know how to set up the bounds for any spherical triple integrals.

Section 13.1 Vector Fields (old book 16.1) 
1) Know the definition for a conservative vector field (i.e. also called a gradient vector field).  F is called a conservative vector field (or a gradient vector field) if F = gradient of f , for some potential function f.
2) Know how to tell whether or not a vector field is conservative by just looking at it (i.e. lab 3).  No curls ----> conservative; Yes curls -----> not conservative.

Sections 13.2 Line Integrals (old book 16.2)
1) Know the definition of work of F along the curve C:
       W = S F *dr
              C
2) Know when you have to evaluate work the long way (i.e. when F is not conservative) and when you can use the fundamental theorem of line integrals.
       
Section 13.3 Line Integrals (old book 16.3) 
1) Know that if any of the following are true, then all of the following are true and if any of the following are false, then all of the following are false:
       i) F is conservative (i.e. F = gradient of f )
      ii)  S F*dr = 0 for any closed path C
           C
      iii)  S F*dr is independent of path
           C
      iv) S F*dr = f(r(b)) - f(r(a))
           C

2) Be able to briefly describe how to find a potential function f given a conservative vector field F either by the method I posted or the book's method.

Section 13.4 Green's Theorem
1) Know the definitions for closed curves and simple curves, and be able to identify whether or not a curve is closed or simple.
2) Know when you can use Green's theorem:
 S Pdx + Qdy = SS (Qx - Py) dA
C                       D
        C must meet the following criteria:
             i) C must be closed and simple
            ii) C must be counter-clockwise (if not then put a negative sign in front of the integral)
           iii) C must be piecewise smooth

3) Know that D is the area inside of the curve C.

Bonus:
Be able to explain or give examples how not knowing the concepts will give you the wrong answer for line integrals or Green's Theorem even if you think you know how to do the problems.