13/9/21 22:02
Taller 4-Hector Palencia
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Student: Hector Palencia
Date: 09/13/21
Instructor: Luis Eduardo Forero
Course: Análisis Multivariado y Funcional
(1)
Assignment: Taller 4
Find the volume of the region bounded by the coordinate planes, the plane , and the
cylinder .
x + y = 3
y + 4z = 9
2 2
The volume is , the integral of over the region D.dz dy dx∫∫∫
D
F(x,y,z) = 1
First, find the z-limits of integration. A line parallel to the z-axis through a typical point in the xy-plane "shadow" enters the
region at and exits through the cylindrical part of the region. Solving for z in and taking only the part of the
equation which pertains to this problem, that is, the part of the cylinder in the first octant, the following is obtained. z .
(x,y)
z = 0 y + 4z = 9
2 2
=9 − y2
2
Thus, the z-limits of the integration are .0 ≤ z ≤ 9 − y2
2
Next, find the y-limits of integration. On the xy-plane, where , a line parallel to the y-axis passing through a typical point
, enters the "shadow" on the xy-plane at and exits through the line . Thus, the y-limits of the integration is
.
z = 0
(x,y) y = 0 y = 3 − x
0 ≤ y ≤ 3 − x
Finally, find the x-limits of integration. Consider the set of all lines parallel to the y-axis that intersect the shadow in the previous
step. The smallest x-value for any line in that set is 0 and the largest x-value for any line in that set is . Thus, the x-limits of
integration are .
3
0 ≤ x ≤ 3
With the information obtained in the previous steps on the limits, the volume integral
.dz dy dx∫∫∫
D
= dz dy dx∫
0
3
∫
0
3 − x
∫
0
9 − y2
/
2
Take the integral with respect to z.
dz dy dx∫
0
3
∫
0
3 − x
∫
0
9 − y2
/
2
= dy dx∫
0
3
∫
0
3 − x
[ ]z 9 − y2
/
2
0
Substitute the upper and lower limits of z and simplify to obtain the following.
dy dx∫
0
3
∫
0
3 − x
[ ]z 9 − y2
/
2
0= dy dx∫
0
3
∫
0
3 − x
9 − y2
2
In order to take the integral with respect to y, trigonometric substitutions must be made. However, if the double-integral version
of Fubini's Theorem is applied to the integration so that the integrals are switched, then the extra y factor will be obtained from
the upper-limit of y when taking the integral with respect to x. This switching of the integrals will simplify the double integration.
Before switching the integrals, the y-limits and x-limits must be identified. On the xy-plane, a line parallel to the x-axis enters the
"shadow" at and exits through a point on the line . Thus, the range of x is x = 0 x + y = 3 0 ≤ x ≤ 3 − y.
To find the y-limits of integration, consider the set of all lines parallel to the x-axis that intersect the shadow in the previous step.
The smallest y-value for any line in that set is 0 and the largest y-value for any line in that set is . Thus, the y-limits of
integration are
3
0 ≤ y ≤ 3.
