The value of the definite integral \(\int_{-3}^{3} \int_{-2}^{2} \int_{-1}^{1}…
2023
The value of the definite integral
\(\int_{-3}^{3} \int_{-2}^{2} \int_{-1}^{1} (4x^2y - z^3) \, dz \, dy \, dx \)
is_______ . (Rounded off to the nearest integer)
Attempted by 47 students.
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Correct answer: 0
Answer: 0
Explanation:
Split the integrand into two parts: 4x^2y and -z^3.
For the term with 4x^2y: the y-integral over [-2, 2] is zero because y is an odd function on a symmetric interval (∫_{-2}^{2} y dy = 0). Therefore the entire contribution from 4x^2y is zero.
For the term with -z^3: the z-integral over [-1, 1] is zero because z^3 is an odd function on a symmetric interval (∫_{-1}^{1} z^3 dz = 0). Hence this term also contributes zero.
Combining both results gives the value of the triple integral as 0.