Which integral represents the area under the curve of the function f(x) = x^2 from x=1 to x=4?

The area under the curve of a function within a specified interval can be determined by calculating the definite integral of the function over that interval. In this case, the function is f(x) = x², and we are interested in the area between x = 1 and x = 4.

The correct integral to represent this area is the one that starts at the lower limit of x = 1 and goes up to the upper limit of x = 4. Thus, the integral ∫ from 1 to 4 of x² dx captures the area under the curve perfectly within these bounds.

When evaluating this integral, you will be finding the exact area under the curve for values of x that lie specifically between 1 and 4, which aligns exactly with the given range.

Other integrals proposed either extend the interval beyond what is needed or start at a point that does not satisfy the requirement of being between 1 and 4. For instance, starting from 0 or going beyond 4 does not reflect the specified area accurately. Hence, the integral defining the area under the curve correctly is ∫ from 1 to 4 x² dx.