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Differential and integral calculus by Feliciano and Uy Differential and integral calculus is a branch of mathematics that deals with the rate of change, commonly represented in terms of the derivative and the integral. It is used in many scientific disciplines such as physics, chemistry, economics, engineering, computer science. Its practical use is to solve problems involving rates of change such as finding the slope at any point on a line when given two endpoints or finding areas under curves when given coordinates for x or y values. This article provides an introduction to differential and integral calculus for students who are unable to take introductory calculus courses due to schedule conflicts or other issues. Differential calculus is used to define the derivative of a function and by this it is possible to find the rate of change of a function's value over time. An example of a function of one variable is given below: The above equation can be written as: Here, we wish to find the rate at which x changes with respect to time. Since the value for y in this equation does not depend on x, it can be ignored when solving for x. Thus, x will be taken as constant throughout the solution process. In order to find the rate of change of y, the derivative is taken with respect to t: This gives us: What we are interested in here is the rate at which y changes with respect to t. The above equation can be written as: Now, if we take the derivative of both sides with respect to t, we will get: Integrating this out will yield: Thus, our formula for finding the rate of change of y with respect to t is given by: As you can see, this formula is identical to that described above. However, it does not involve any differentiation at all. This allows us to find the rate of change even when the variables are not independent. Integral calculus is used to find areas or volumes under curves. An example of an integral is shown below: This equation can be written as: If we want to know the area between y = f(x) and the x-axis, we can integrate this using u to be x, yielding: This area under the curve is equal to the area bounded by the vertical lines x = a and x = b, which can be calculated by subtracting the areas of these two rectangles from that of an outer rectangle with base b and height h. This area under the curve is equal to ab. It can be shown that this formula is equivalent to an explicit version of the formula for the volume under a parabola given above by integrating out f(x). The following article is about how to find the integral of a curl of a function. This article will give you some examples on how to do this using some basic rules that are easy to remember. We are also going to show you some quick shortcuts that may come in handy when working with integrals. cfa1e77820
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