The region bounded by $$ y=x^{3} $$, the $$ x $$-axis, the line $$ x=1 $$, and $$ x=5 $$

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1. Identify the curves and boundaries of the region:
   - The curve $$ y=x^3 $$.
   - The $$ x $$-axis (which is $$ y=0 $$).
   - The vertical lines $$ x=1 $$ and $$ x=5 $$.

2. Since for $$ x $$ between 1 and 5 the curve $$ y=x^3 $$ is above the $$ x $$-axis, the area of the region is given by the integral of $$ y=x^3 $$ from $$ x=1 $$ to $$ x=5 $$.

3. Set up the integral for the area $$ A $$:
   $$
   A=\int_{1}^{5}x^3\,dx
   $$

4. Evaluate the integral:
   - Compute the antiderivative of $$ x^3 $$:
     $$
     \int x^3\,dx=\frac{x^4}{4}+C
     $$
   - Apply the limits:
     $$
     A=\left[\frac{x^4}{4}\right]_{x=1}^{5}=\frac{5^4}{4}-\frac{1^4}{4}
     $$

5. Compute the numerical values:
   - Calculate $$ 5^4=625 $$ and $$ 1^4=1 $$:
     $$
     A=\frac{625-1}{4}=\frac{624}{4}=156
     $$

6. Thus, the area of the region is:
   $$
   A=156
   $$
The area of the region is 156.

The region bounded by , the -axis, the line , and

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