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].
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.
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.
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.
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.
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.