To estimate area enclosed between the x-axis and a curve on a certain bounded region you can use rectangles or parallelograms.
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Integrate between the bounds.
Calculus can be used to find the surface area of any object given that you know the equation describing said object. It's usually easier to find the area from experiment or through using a combination of existing models to approximate the surface area
Such as? If you can break the shape up into triangles you can find the area that way. Or, you can get into calculus-based equations if you have an equation for the random shape.
Here's an example calculus question: Find lim (x^2-4)/(x^2+2x-8) using l'hopital's rule. x->2
Calculus has been beneficent in nearly all areas of human life and has provided the world with numerous technological advancements such as the ability to measure the area of a piece of property without having to manually do so (e.g., instead of building and measuring a pool to find its area, all we must do is calculate an estimate using calculus). It provides us an exact method to find the rate of change of a function, allows us to easily prove theories and formulas, and allows us to calculate volumes, optimums, profits, and much more by using only functions and graphs.