Integration to Find Area Under a Curve

Definition

Integration is a process associated with finding the area under a curve.
The definite integral of a function can be interpreted as the area under the graph of the function and above the x-axis, bounded by two vertical lines.

To utilize integration for finding areas, we identify the function that represents the curve, decide the limits of the integration which are the x-coordinates of the endpoints along the x-axis, and use the Fundamental Theorem of Calculus to evaluate the definite integral.
We use definite integrals to find the net area between the function and the x-axis from a to b.
To find the total area under a curve where the function dips below the x-axis, we must split the integral at the x-coordinates where the function crosses the x-axis, and take the absolute value of each integral.

Start by identifying the integral. For instance, if you have the function f(x) = x^2 from 0 to 2, the integral would be ∫f(x) dx from 0 to 2.
Next, find the antiderivative F(x) of f(x). In this case, the antiderivative of x^2 is (1/3)x^3.
Now, apply the Fundamental Theorem of Calculus, which in basic terms means subtract the antiderivative evaluated at the lower limit from the antiderivative evaluated at the upper limit. That is, F(2)-(F(0)) = (1/3)2^3 - (1/3)0^3 = 8/3 units^2 which is the area under the curve.

Neglecting to take the absolute value of each integral when the function falls below the x-axis.
Forgetting to subtract the lower limit value from the upper limit value after finding the antiderivative.

The fundamental application of integration is to calculate the area under a curve, which is crucial in physical sciences and engineering such as in determining the displacement of an object given its velocity function.
Other applications include calculating work done, finding centroid, moment of inertia and much more. Many economic and statistical models also rely on integration.