I will explain what is Fundamental Theorem of Calculus in this reflection. Let a function f(x) be continuous on an interval from a to b, and F(x) be the indefinite integral of f(x). If F(x) is differentiable, then the derivative of F(x) will equal to f(x). Let G(x) be the antiderivative of f(x), then the integral of f(t) from r to l will equal to G(r) minus G(l).

With this knowledge, we can calculate area between functions and axises or between functions and functions. For example, if we want to calculate the area above the x-axis and under the function y = 2x − x^2, we should calculate the antiderivative of y first which is x^2 − 1/3(x^3). Then we need to calculate points when function y and x-axis intersect with each other which are  0 and 2. By Fundamental Theorem of Calculus, we plug x=2 and x=0 into y’s antiderivative x^2 − 1/3(x^3) to get the final answer 4/3.