I hope you don't give your satisfactory answer to others before I finish typing. I had such a painful experience once. I said it was painful because I typed hundreds of words and symbols and found that I couldn't pick them up when I answered them. Ok, I'll answer your question briefly.

First of all, we should understand the significance and purpose of learning definite integral: for practical application. Then let's explain it one by one from the application of definite integral, as follows:

1. definite integral can be used to find the distance of variable-speed linear motion:

V=V(t) is a function of time interval (T 1, T2), and generally V(t) is greater than or equal to zero. Here, we can easily find out the moving distance of an object in T 1 and T2 by using definite integral. Remember here that v is the y axis and t is the x axis.

2. The definite integral can be used to find the area of the graph, but remember that the function of the definite integral is to find the area of the graph surrounded by the curve and the X axis or the Y axis. When calculating the area of a graph, we need to solve the graph by sections, instead of saying that a complete definite integral must be the area of the graph (this understanding is completely wrong). Definite integral is only a tool that is partially complete and wholly incomplete. Here is an example:

The graphic area enclosed by parabola y*y=2x and straight line y=x-4. Here, we choose the ordinate y as the integral variable, and its variation interval is, dA=(y+4-y*y/2)dy, take (y+4-y*y/2)dy as the integrand expression, and definite the integral in the closed interval, then the required area is 18. (Consider taking the abscissa x as an integral variable.

Find the graphic area enclosed by the ellipse x*X/(a*a)+y*y/(b*b)= 1 This is an ellipse stacked about two coordinate axes. Let the area of the ellipse be a and the area of the ellipse in the interval 1 be A 1, then A=4*A 1. We can use the parametric equation of ellipse to find out the area of A 1 first, and then multiply it by 4 to get it.

3. Find the volume of the rotating body, the arc length of the plane curve and the work done by the variable force along the straight line. I won't list them here.

What I want to tell you is that definite integral is only for the convenience of finding the area of the graph enclosed between the variable curve and a certain coordinate axis. As for complex graphics, you need to divide the graphics into several parts, then find them separately and combine them to get the total area. Definite integral is not to find the overall area of the graph, but to find the area of the graph enclosed between the variable curve and a coordinate axis conveniently! Don't exaggerate its function! Equations are dead and people are alive.

Oh, as for the problem of finding a straight line by definite integral, it can only be used for problems such as stocks. It is used to average a probability problem. Set a constant line of moving average, above which it is safe and below which it is unsafe. How many days should this safety line be set? The upper area and the lower area of the stock price chart just cancel each other out. You can also set this line higher, and the actual application is much more complicated according to the safety needs. Well, you're still in the primary stage, so you shouldn't be able to use it for the time being. Definite integral has different meanings for different problems!