Consider the function f(x) = x 2 The goal is to find a general formula for the area under this curve on the interval [0, b]. (a) Start by looking at finding the area under the graph of f(x) on [0, 1]...

Consider the function f(x) = x²

The goal is to find a general formula for the area under this curve on
the interval [0, b].

(a) Start by looking at finding the area under the graph of f(x) on [0, 1] using N rectangles, using the right endpoints as the test points. What do the endpoints look like? Use this to write the approximate area in summation notation.

(b) What if this is done on [0, 2], still using N rectangles? How does this change the formulas and the sum? What are the right endpoints here?

(c) What if this also done on [0, b] for some positive number b using N rectangles? What are the endpoints here?

(d) With this last sum, expand things out and use the power sum rules to get this to an expression only depending on N.

(e) Take the limit as N → ∞ to get the area under the graph of f(x) over [0, b].