The integral is usually called the anti-derivative , because integrating is the reverse process of differentiating. The fundamental theorem of calculus shows that antidifferentiation is the same as integration. The physical concept of the integral is similar to the derivative. For the derivative, the motivation was to find the velocity at any point in time given the position of an object. Just as the derivative gave the instantaneous rate of change, the integral will give the total distance at any given time.
The integral comes from not only trying to find the inverse process of taking the derivative, but trying to solve the area problem as well. Just as the process of differentiation is used to find the slope at any point on the graph, the process of integration finds the area of the curve up to any point on the graph. Before integration was developed, we could only really approximate the area of functions by dividing the space into rectangles and adding the areas.
We can approximate the area to the x axis by increasing the number of rectangles under the curve. The area of these rectangles was calculated by multiplying length times width, or y times the change in x.
After the area was calculated, the summation of the rectangles would approximate the area. As the number of rectangles gets larger, the better the approximation will be. This is formula for the Riemann Summation, where i is any starting x value and n is the number of rectangles:. This was a tedious process and never gave the exact area for the curve. Luckily, Newton and Leibniz developed the method of integration that enabled them to find the exact area of the curve at any point.
Similar to the way the process of differentiation finds the function of the slope as the distance between two points get infinitesimally small, the process of integration finds the area under the curve as the number of partitions of rectangles under the curve gets infinitely large. Generally, an integral assigns numbers to functions in a way that can describe displacement, area, volume and even probability. This type of integral relates to numerical values.
It is used in pure mathematics, applied mathematics, statistics, science and many more. However, the very basic concept of a definite integral describes areas. The definite integral of a function f over an interval [a,b] represents the area defined by the function and the x-axis from point a to point b , as seen below.
You might be wondering what "d"x means. Formally, it doesn't mean anything but rather it tells you which variable you are differentiating with respect to or in our case, tells you the variable of integration. When we say the area defined by the function f with the x-axis, we mean the net area. The net area is not the same as absolute area.
If the graph of the function is above the x-axis, then it is said that the net area is positive. If it is below, the net area is negative. This might be harder to grasp at first. This is visualised below:. At the same time, the video describes Riemann sums. These are used to compute integrals. Generally, the Riemann sum of a function phi is. While this is usually simpler, it might not be easiest or fastest way to compute integrals.
There are many, many ways different formulas for integrals, which I won't cover in this answer. Hydrostatic force is only one of the many applications of definite integrals we explore in this chapter.
From geometric applications such as surface area and volume, to physical applications such as mass and work, to growth and decay models, definite integrals are a powerful tool to help us understand and model the world around us. When Lake Mead, the reservoir behind the dam, is full, the dam withstands a great deal of force.
However, water levels in the lake vary considerably as a result of droughts and varying water demands. To find the area between two curves defined by functions, integrate the difference of the functions.
If the graphs of the functions cross, or if the region is complex, use the absolute value of the difference of the functions. In this case, it may be necessary to evaluate two or more integrals. We consider three approaches—slicing, disks, and washers—for finding these volumes, depending on the characteristics of the solid.
We can use this method on the same kinds of solids as the disk method or the washer method; however, with the disk and washer methods, we integrate along the coordinate axis parallel to the axis of revolution.
With the method of cylindrical shells, we integrate along the coordinate axis perpendicular to the axis of revolution.
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