integration review

Integration review:

Defintions:  Know these and be able to use them.  

Riemann sum

Double integral

Area and volume as a double integral

Iterated integral

Type 1 and type 2 regions

Average value of a function on a region

first and second moments (about a line) of a function on a region

center of mass of a lamina, centroid of a region

Radius of gyration of a lamina about the y-axis (x-axis)

Triple integral

Volume as a triple integral

Line integrals with respect to arc length, x, y, or z (if the curve is a space curve)

Length as a line integral

vector field  Force field

Work as a line integral

gradient field

conservative vector field

potential

Theorems:  Know these their consequences

Linearity of double integrals (and triple integrals and line integrals).

Comparability of double integrals ( and triple integrals and line integrals)

Fubini's Theorem

Change of variables Theorem

Fundamental Theorem for line integrals

Characterization Theorem for conservative vector fields

Green's Theorem

Typical problems

16_12
Compute a Riemann sum for a function
Compute a Riemann sum for a table of values
Double integral as the limit of a Riemann sum.
Compute a double integral by realizing it as a volume
Use Fubini's Theorem to compute a double integral
Compute an iterated integral
Compute a volume using an iterated integral
Compute an average height
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16_34
double integral trivia
compute a double integral over a type 1 or type 2
another compute a double integral over a type 1 or type 2
another compute a double integral over a type 1 or type 2
reverse the order of integration
convert to polar coordinates
compute a volume using polar coordinates
What's left after a hole is drilled in a sphere?
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16_56
compute the center of mass of a triangular region with variable density
Find center of mass of quarter disk
Find radius of gyration of rectangle in first quadrant
compute the area of a plane which lies above a given triangle.
another compute a double integral over a type 1 or type 2
compute an area using polar coordinates
What's left after a hole is drilled in a sphere?
>
16_78
evaluate a triple iterated integral
compute an average temperature
compute an average temperature using cylindrical coordinates
compute a mass using spherical coordinates
y-coordinate of the center of mass.
Find average density of a prism
change of variable problem for double integrals
change of variable problem for triple integrals

17_12
identify a gradient field
Find the area of a fence.  Find its average height
Find the mass and center of mass of a wire  Find its average density.
Calculate line integrals with respect to x or y
Calculate a work integral along various paths from A to B
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17_34
Calculate line integrals for gradient fields
Find potentials.

Calculate work integral using Green's theorem

Express area as line integral and evaluate.

Sample questions.  More later in week .

1.   Draw a picture of the region R for the integral    .  Then reverse the order of integration and evaluate.

2.   Find the average value of      over the  rectangular solid with opposite corners  (0,0,0) and (1,2,3)

3.  Let a > 0. Let  be the solid below the plane , above the xy-plane and inside the cylinder .  Express the volume of  as an iterated double integral in polar coordinates.  Evaluate. Calculate the centroid of .

4.  Find the maximum and minimum values M and m of  on the domain D = {(x,y)|   }.   Then show that  the integral of f over D  is between    and

5.  Find the center of mass of the line segment  from (0,1,0) to (2,0,1)  if the density at each point on the wire is given by .

6.  Show that the vector field  F =<P,Q> = < >  is conservative.   Use this to evaluate  the work integral   .  over any nice curve C from (0,0) to (1,1).

7. Obtain the formula for the area of a triangle with base b and height h by evaluating a line integral