The area problem give us one of the interpretations of a definite integral that will lead us to the definition of the definite integral.
Let the Limit of n approaches to infinity (n → ∞)
Reviewing what you have learned in the topic Area Problem and the Riemann Sum series. As long as f is continuous the value of the limit is independent of the sample points xi used. If we let the Limit of n approaches to infinity (n → ∞). The definite integral of f from a to b is the define
provided the limit exists. The definite integral is defined to be exactly the limit and summation that we looked at in the previous topic to find the net area between a function and the xaxis.
Area Under a Curve
The area between the graph of y = f(x)and the xaxis is given by the definite integral below. This formula gives a positive result for a graph above the xaxis, and a negative result for a graph below the xaxis.
Note: If the graph of y = f(x) is partly above and partly below the xaxis, the formula given below generates the net area. That is, the area above the axis minus the area below the axis.
Formula: 

Example 1:  Find the area between y = 7 – x2 and the xaxis between the values x = –1 and x = 2.

Example 2:  Find the net area between y = sin x and the xaxis between the values x = 0 and x = 2π.

Area Between a Curve
The area between curves is given by the formulas below.
Formula 1:  
for a region bounded above and below by y = f(x) and y = g(x), and on the left and right by x = a and x = b.


Formula 2:  
for a region bounded left and right by x = f(y) and x = g(y), and above and below by y = c and y = d.


Example 1:1  Find the area between y = x and y = x2 from x = 0 to x = 1. 


Example 2:1  Find the area between x = y + 3 and x = y2 from y = –1 to y = 1. 
Continue and apply what you have learned by answering some problems that you can find on this link Area Under a Curve and Between a Curve – Set 1 Problems.
credit: Bruce Simmons, Paul Dawkins (Lamar University)©2013 www.PinoyBIX.com