Double Integrals over Rectangles

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Section 16.1

• Double Integrals over Rectangles

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A CLOSED RECTANGLE
A closed rectangle in the plane is the region
given by
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VOLUME OVER A RECTANGLE
Consider the function f defined on a closed
rectangle R and that f (x, y) 0. The graph of
f is a surface with equation z f (x, y).
Let S be the solid that lies above R and under
the graph of f, that is Our goal is to find the
volume of the solid S.
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PARTITIONING THE RECTANGLE

• Partition a, b into m equal subintervals xi -
  1, xi of equal width ?x (b - a)/m.
• Partition c, d into n equal subintervals yj -
  1, yj of equal width ?y (c - d)/n.
• Create subrectangles Rij, each of area ?A ?x
  ?y as follows

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VOLUME OVER A RECTANGLE
To find the volume over a close rectangle R

• Choose a sample point in each Rij.
• Find the volume in the rectangular column formed
  by Rij, that is
• Sum all the volumes and take the limit.

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THE DOUBLE INTEGRAL
The double integral of f over the rectangle R
is if the limit exits.
NOTE The function f does not have to be
positive. Its graph can be below the xy-plane.
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For each subrectangle Rij, if we choose the
sample point to be (xi, yj), then the expression
for the double integral simplifies to
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VOLUME AND DOUBLE INTEGRALS
If f (x, y) 0, then the volume V of the solid
that lies above the rectangle R and below the
surface z f (x, y) is
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DOUBLE RIEMANN SUM
The sum is called the double Riemann sum and is
used to approximate the value of the double
integral.
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MIDPOINT RULE FOR DOUBLE INTEGRALS
where is the midpoint of xi - 1, xi
and is the midpoint of yj - 1, yj.
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AVERAGE VALUE
The average value of a function f of two
variables defined on a rectangle R is where
A(R) is the area of R.
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PROPERTIES OF THE DOUBLE INTEGRAL
If f (x, y) g(x, y) for all (x, y) in R, then