Generally.
Area = the definite integration from a to b [ f(x) - g(X)]dx
Say you have two functions( y = e^x and y = x) and you want to find the area between them on an interval ( say 1 to 3 ) So you set the top function(e^x) subtracting the bottom function(x) and integrate them, Insert the values, b - a in the integrated functions and get the value of the area.
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If the values of the function are all positive, then the integral IS the area under the curve.
Differential calculus is concerned with finding the slope of a curve at different points. Integral calculus is concerned with finding the area under a curve.
Do a line integral.
Finding the area under a curve or the length of a line segment. These are real life uses, not just fun in your math class.
The distance covered between two points in time is the area under the graph between the two points.