Definition: The area $A$ of the region $S$ that lies under the graph of the continuous function $f$ is the limit of the sum of the areas of approximating rectangles (a) Use the above definition to determine which of the following expressions represents the area under the graph of $f(x)=x^3$ from $x = 0$ to $x = 1$.

(b) Evaluate the limit that is the correct answer to part (a). You may find the following formula for the sum of cubes helpful:

The value of the limit is .

You can earn partial credit on this problem.