Question

The value of the definite integral \(\displaystyle \int_{-3}^{3}\int_{-2}^{2}\int_{-1}^{1}\big(4x^{2}y - z^{3}\big)\,dz\,dy\,dx\) is ________. (Rounded off to the nearest integer)

Hint:

Before grinding through multivariable integrals, check for odd/even symmetry over symmetric limits. Odd parts vanish immediately, saving time.

Answer 1

Step 1: Observe the nature of the integrand

The integrand is:

4x²y − z³

This expression can be split into two separate terms:

• 4x²y → depends on y
• −z³ → depends on z

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Step 2: Use symmetry over the limits

Consider the limits of integration:

• y ∈ [−2, 2]
• z ∈ [−1, 1]

Now observe:

• 4x²y is an odd function in y
• z³ is an odd function in z

An odd function integrated over symmetric limits evaluates to zero.

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Step 3: Apply this to the triple integral

Since:

• The y-dependent part integrates to 0 over [−2, 2]
• The z-dependent part integrates to 0 over [−1, 1]

The contribution of the entire integrand over the given rectangular region is zero, regardless of the x-limits.

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Final Answer:

0