Integral calculus is the mathematics of integration, which is used to calculate volumes and areas of complicated shapes. Whereas a rectangle or circle has a simple mathematic function for finding its area, an undefined, squiggly shape is not as straightforward, and must be integrated. This brief overview of integral calculus serves to explain how integrals are used to perform these calculations. For the purposes of this overview, only the calculation of areas will be explained. Integral calculus can also be used to calculate volumes, but the math becomes more complicated.
Integrals are used to find the area under a curve on a graph, where the curve is represented by an algebraic function, such as "x + 2". The area is bound at the top by the function, at the bottom by the x-axis, and on the sides by whatever boundaries you choose, which we shall call endpoints. Performing the integral function from the left endpoint to the right endpoint is the mathematical way of computing the area inside of the boundaries. An integral is an exact calculation of the specified area, but to understand the concept, it is helpful to start with an approximation.
Imagine that the area underneath a function on a graph is divided into rectangles of equal width, but with heights equal to their coinciding points on the function. From the left endpoint to the right endpoint, you would see vertical rectangles, each of equal width, but with varying heights. The top of this picture would look almost like the function itself, except rigid, rectangular corners replace the formerly smooth curve. Could you calculate the area of this new, approximated shape? If you know that the area of a rectangle is equal to the height multiplied by the width, and you could measure the height and width of each rectangle, then you could calculate the area of each rectangle and add them all up. The total area would then be the area of your shape.
Put simply, an integral function is a way to divide up the area under a function, and then add them all up, except that the width of each rectangle is so narrow that it "approaches zero" and the number of total rectangles is so large that it "approaches infinity".
Integral calculus, then, is the mathematics of integration. The curve, normally represented as an algebraic function dependent upon variables (such as "x" in the earlier example), can be integrated. The integral, which is the area, is reported as a new function, dependent upon the same variables. If the values of the variables are known, they can then be plugged into the equation, and the area under the curve calculated.