Yes, this can be done exactly with calculus (and is probably intended to lead you in the direction of some calculus concepts); but they only want an estimate, and they tell you that you can do different things to get different levels of accuracy.

What you did is a very good first try; but where does the 4 come from?

One way you can improve this is to find one set of squares that is entirely contained within the figure, and another that contains the figure. (That is, one is below and the other is above.) Then you can be sure the the actual area is between the two numbers.

Another thing you could do is to find a figure made of straight lines that is as close as you can come to the actual figure. You might draw lines between the points on the curve, and then add up rectangles and triangles, or trapezoids, to find the area, which will be a closer overestimate.

See what you can do; as they say, it's meant to be a "critical thinking activity", and an opportunity to "be creative", so just have fun with it!