Draw a series of rectangles under the curve, covering the area you want to find. The rectangles set on the X axis and the height of each rectangle is determined by the curve you are using. At a point Xp on the X axis, the height of the rectangle is the value you get if you plug Xp into the function and solve. For example, if you are computing the area under the curve Y = X^2 + 1 between X = 1 and X = 10, the height of the rectangle at the point X = 5 is Y = 5^2 + 1 = 26.
Add up the areas of all the rectangles to approximate the area under the curve. The thinner the rectangles, the more accurate your estimate will be, because the tops of the rectangles do not match the curve exactly. For example, if the curve is constantly decreasing over the target interval, and the curve intersects the rectangle at the right-hand corner, the rectangle is completely under the curve and the approximation of the area will be low. Conversely, the approximations are sometimes high.
Look at what happens when the width of the rectangles goes to zero. Fermat developed an algebraic expression for the sum of the areas of the rectangles. If the width of the rectangles actually were zero the area under under the curve was the sum of an infinite number of zeros -- a confusing concept at best. If we look at the limit of this process, however, we can come up with a realistic answer. For example, look at what happens to Y = (X^2 - 1)/(X - 1) when X gets closer and closer to 1. At the point when X = 1, Y is undefined but you can see, by checking a few values close to X, that as X approaches 1, Y approaches 2.