Problems: Regions of Integration
1. Find the mass of the region R bounded by
y = x + 1; y = x2 ; x = 0 and x = 1, if density = δ(x, y) = xy.
y = x + 1
Answer:
Inner limits: y from x2 to x + 1. Outer limits: x from 0 to 1.
x+1
1
⇒
M =
δ(x, y) dA =
xy dy dx
R
x=0
R
y=x2
y = x2
1
[
x+1
y2 [
x(x + 1)2 x5
x3
x x5
xy dy =
x
[[
=
−
=
+ x2
+
−
.
2
x2
2
2
2
2
2
[1
3
5
4
3
2
6
x
x x
x
x
x
x
[
1 1 1
1
5
+ x
2 +
−
dx =
+
+
−
[[
= + +
−
= .
2
2
2
8
3
4
12
0 8 3 4 12
8
x+1
Inner:
x2
1
Outer:
0
Note: The syntax y = x2 in limits is redundant but useful. We know it must be y because
of the dy matching the integral sign.
2. Find the volume of the tetrahedron shown below.
1 z = f (x, y)
�
1
1
Tetrahedron
Answer: The surface has height: z = 1 − x − y.
y
1
y =1−x
R
1 x
Region R
1
1−x
1 − x − y dy dx.
Limits: inner: 0 < y < 1 − x, outer: 0 < x < 1. ⇒ V =
x=0
1−x
Inner:
y=0
1
Outer:
0
y=0
[1−x
y 2 [[
1
x2
1 − x − y dy = y − xy −
[
= 1 − x − x + x2
− + x −
.
2 0
2
2
1
x2
1 1 1
1
−x+
dx = − + = .
2
2
2 2 6
6
1
MIT OpenCourseWare
http://ocw.mit.edu
18.02SC Multivariable Calculus
Fall 2010
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