To evaluate the integral [tex]\(\int_4^4 x^3 \, dx\)[/tex], let's carefully consider the mathematical principles involved in evaluating definite integrals.
### Step-by-Step Solution:
1. Understanding Definite Integrals:
A definite integral [tex]\(\int_a^b f(x) \, dx\)[/tex] represents the net area under the curve [tex]\(f(x)\)[/tex] from [tex]\(x = a\)[/tex] to [tex]\(x = b\)[/tex].
2. Special Case: Integrals with the Same Upper and Lower Limit:
When the upper and lower limits of the integral are the same, i.e., [tex]\(a = b\)[/tex], the integral represents the area under the curve from a point to the same point. Geometrically, there is no width to this region, so the area under the curve is zero.
### Applying the Special Case:
Given the integral [tex]\(\int_4^4 x^3 \, dx\)[/tex], notice that the upper and lower limits are both 4.
[tex]\[
\int_4^4 x^3 \, dx
\][/tex]
By the geometric interpretation of definite integrals with the same upper and lower limit, we can conclude that:
[tex]\[
\int_4^4 x^3 \, dx = 0
\][/tex]
### Conclusion:
The value of the integral [tex]\(\int_4^4 x^3 \, dx\)[/tex] is [tex]\(\boxed{0}\)[/tex].
By following the principles of definite integrals, we understand that anytime the upper and lower limits are identical, the integral is equal to zero, regardless of the integrand [tex]\(f(x)\)[/tex].
