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)

Question

The integer value printed by the ANSI-C program given below is ________.


int funcp(){
    static int x = 1;
    x++;
    return x;
}

int main(){
    int x,y;
    x = funcp();
    y = funcp() + x;
    printf("%d\n", (x+y));
    return 0;
}

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

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.

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.

Final Answer:

0

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)

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Solution and Explanation

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