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
Evaluate the integral . x + y + z dz dy dx∫
0
4
∫
0
3
∫
0
2
222
Evaluate the triple integral using the same method used to evaluate a double integral. The only difference is one more
integration step will be needed to obtain the final value.
Begin by integrating with respect to z.
dy dx∫
0
4
∫
0
3
∫
0
2
x2+ y2+ z2 dz = dy dx ∫
0
4
∫
0
3
x2z + y2z + z3
3
2
0
Substitute the upper and lower limits of z and simplify the resulting expression.
dy dx ∫
0
4
∫
0
3
x2z + y2z + z3
3
2
0
= 2x + 2y + dy dx∫
0
4
∫
0
3
2 2 8
3
Next integrate with respect to y.
dx∫
0
4
∫
0
3
2x2+ 2y2+8
3dy = dx∫
0
4
2x2y + 2
3y3+8
3y
3
0
Substitute the upper and lower limits of y and simplify the resulting expression.
dx∫
0
4
2x2y + 2
3y3+8
3y
3
0
= 6x + 26 dx∫
0
4
2
Finally integrate with respect to x.
6x + 26 dx∫
0
4
2=
2x3+ 26x
4
0
Substitute the upper and lower limits of x to obtain the value of the triple integral.
2x3+ 26x
4
0= 232
Thus, . x + y + z dz dy dx∫
0
4
∫
0
3
∫
0
2
222 = 232