2. You may recall that we can do a Riemann sum to
approximate the area under the graph of a function
of one variable by adding the areas of the
rectangles that form below the graph resulting from
small increments of x (x) within a given interval
[a, b]:
x
y
y= f(x)
a b
x
3. z
Similarly, it is possible to obtain an approximation of
the volume of the solid under the graph of a
function of two variables.
y
x
a
b
c d
R
y= f(x, y)
4. a. Suppose g1(x) and g2(x) are continuous functions on [a, b] and
the region R is defined by R = {(x, y)| g1(x) y g2(x); a x b}.
Then,
2
1
( )
( )
( , ) ( , )
b g x
R a g x
f x y dA f x y dy dx
x
y
a b
y = g1(x)
y = g2(x)
R
5. Let R be a region in the xy-plane and let f
be continuous and nonnegative on R.
Then, the volume of the solid under a
surface bounded above by z = f(x, y) and
below by R is given by
( , )
R
V f x y dA
6. z
To find the volume of the solid under the surface, we
can perform a Riemann sum of the volume Si of
parallelepipeds with base Ri = x ☓ y and height
f(xi, yi):
y
x
a
b
c d
x
y
z = f(x,
y)
R
7. z
To find the volume of the solid under the surface, we
can perform a Riemann sum of the volume Si of
parallelepipeds with base Ri = x ☓ y and height
f(xi, yi):
y
x
a
b
c d
x
y
Si
z = f(x,
y)
R
8. y
a
b
c d
z
x
To find the volume of the solid under the surface, we
can perform a Riemann sum of the volume Si of
parallelepipeds with base Ri = x ☓ y and height
f(xi, yi):
z = f(x,
y)
9. z
The limit of the Riemann sum obtained when the number of
rectangles m along the x-axis, and the number of
subdivisions n along the y-axis tends to infinity is the value
of the double integral of f(x, y) over the region R and is
denoted by
z = f(x,
y)
y
x
a
b
c d
x
y
R
( , )
R
f x y dA
y ·n
x
·m
10. b. Suppose h1(y) and h2(y) are continuous functions on [c, d]
and the region R is defined by R = {(x, y)| h1(y) x
h2(y); c y d}.
Then, 2
1
( )
( )
( , ) ( , )
d h y
R c h y
f x y dA f x y dx dy
x
y
c
d
x = h1(y) x = h2(y)
R
11. 4
3
2
1
1 2 3 4
Evaluate ∫R∫f(x, y)dA given that f(x, y) = x2 + y2 and
R is the region bounded by the graphs of g1(x) = x
and g2(x) = 2x for 0 x 2.
Solution
The region under consideration is:
x
g2(x) = 2x
R
y
g1(x) = x
Example 2, page 593
12. Evaluate ∫R∫f(x, y)dA given that f(x, y) = x2 + y2 and
R is the region bounded by the graphs of g1(x) = x
and g2(x) = 2x for 0 x 2.
Solution
Using Theorem 1, we find:
2 2
2 2
0
( , ) ( )
x
R x
f x y dA x y dy dx
2
2
2 3
0
1
3
x
x
x y y dx
2
3 3 3 3
0
8 1
2
3 3
x x x x dx
2
3
0
10
3
x dx
2
4
0
5
6
x
1
3
13
Example 2, page 593
13. 1
1
Evaluate ∫R∫f(x, y)dA, where f(x, y) = xey and R is the
plane region bounded by the graphs of y = x2 and y = x.
Solution
The region under consideration is:
x
g1(x) = x2
R
y
g2(x) = x
The points of intersection of
the two curves are found by
solving the equation x2 = x,
giving x = 0 and x = 1.
Example 3, page 593
14. Evaluate ∫R∫f(x, y)dA, where f(x, y) = xey and R is
the plane region bounded by the graphs of y = x2
and y = x.
Solution
Using Theorem 1, we find:
2
1
0
( , )
x
y
R x
f x y dA xe dy dx
2
1
0
x
y
x
xe dx
2
1
0
( )
x x
xe xe dx
2
1 1
0 0
x x
xe dx xe dx
2
1
0
1
( 1)
2
x x
x e e
1 1 1
1 (3 )
2 2 2
e e
Integrating by parts on
the right-hand side
Example 3, page 593
15. Find the volume of the solid bounded above by the
plane z = f(x, y) = y and below by the plane
region R defined by
Solution
The graph of the region R is:
1
1
x
y
2
1
y x
R
Observe that f(x, y) = y > 0 for (x, y) ∈ R.
2
1 (0 1)
y x x
Example 4, page 594
16. Find the volume of the solid bounded above by the
plane z = f(x, y) = y and below by the plane
region R defined by
Solution
Therefore, the required volume is given by
2
1 1
0 0
x
R
V ydA ydy dx
2
1
1
2
0
0
1
2
x
y dx
1
2
0
1
(1 )
2
x dx
1
3
0
1 1
2 3
x x
1
3
2
1 (0 1)
y x x
Example 4, page 594
17. Find the volume of the solid bounded above by the
plane z = f(x, y) = y and below by the plane region
R defined by
Solution
2
1 (0 1)
y x x
z
x
y
The graph of the solid
in question is:
2
1
y x
R
z = f(x, y) = y
Example 4, page 594