Website title: Formal Relation Between Integrals of y=f(x) and y=x

In summary, there is a formal relation between the indefinite integrals \int yxdx and \int xydy, as well as the definite integrals \int_{a}^{b}yxdx and \int_{a}^{b}xydy, where y=f(x) over the interval x\in\left[a,b\right]. However, for the definite integrals, it is important to take note of the differences in the limits, as they refer to x for the dx integral and y for the dy integral.
  • #1
JulieK
50
0
Is there a formal relation that links
[itex]\int yxdx[/itex] OR [itex]\int_{a}^{b}yxdx[/itex]
with
[itex]\int xydy[/itex] OR [itex]\int_{a}^{b}xydy[/itex]
where [itex]y=f(x)[/itex] over the interval [itex]x\in\left[a,b\right][/itex].
 
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  • #2
JulieK said:
Is there a formal relation that links
[itex]\int yxdx[/itex] OR [itex]\int_{a}^{b}yxdx[/itex]
with
[itex]\int xydy[/itex] OR [itex]\int_{a}^{b}xydy[/itex]
where [itex]y=f(x)[/itex] over the interval [itex]x\in\left[a,b\right][/itex].

For the indefinite integral (assuming everything is nice):
[itex]\int xydy =\int xyy'dx[/itex]

For the definite integral there is a a similar relationship, but be careful with the limits. For the dx integral, the limits refer to x, while for the dy integral, the limits refer to y.
 

FAQ: Website title: Formal Relation Between Integrals of y=f(x) and y=x

What is the relation between integrals and derivatives?

The relation between integrals and derivatives is that they are inverse operations. The derivative of a function represents the slope or rate of change of that function at any given point, while the integral of a function represents the cumulative sum of the function's values over a given interval. In other words, the derivative finds the rate of change at a specific point, while the integral finds the total change over an interval.

How are definite and indefinite integrals related?

Definite and indefinite integrals are both types of integrals, but they differ in their notation and purpose. An indefinite integral, also known as an antiderivative, is the inverse operation of a derivative and is used to find the original function from its derivative. A definite integral, on the other hand, has upper and lower limits and is used to find the area under a curve or the net change of a function over a given interval. A definite integral can be found by using the antiderivative and evaluating it at the upper and lower limits of integration.

How do integrals relate to the area under a curve?

Integrals are used to find the area under a curve by calculating the sum of infinitesimally small rectangles that approximate the shape of the curve. This is known as the Riemann sum, and as the width of the rectangles approaches zero, the sum approaches the exact area under the curve. Therefore, the integral of a function represents the exact area under the curve between two points on the x-axis.

What is the fundamental theorem of calculus?

The fundamental theorem of calculus states that integration and differentiation are inverse operations. This means that if a function f(x) is integrated to obtain F(x), then the derivative of F(x) will be equal to f(x). In other words, the integral and derivative of a function cancel each other out, leaving the original function.

How are integrals used in real-world applications?

Integrals have numerous real-world applications, particularly in physics and engineering. They are used to calculate the area under a velocity-time graph to determine displacement, as well as to calculate the total work done by a force. In economics, integrals are used to calculate the total profit or revenue of a business over a given time period. In addition, integrals are also used in probability and statistics to find the area under a probability distribution curve and calculate the probability of certain events occurring.

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