Second and Higher Order Integrals?

Question

I took an introductory calculus class last semester and we got up to the point of integration. We just covered basic integration but didn't get to integration by parts. I know that you can use integration to determine the area of a curve or the length of a line etc however we didn't cover higher order integrals.

What information does solving higher order integrals give you? What would be the use for them?

Answer 1

Consider the two-dimensional case first. Suppose we have a function of two variables - f(x,y) and we compute its integral over a two-dimensional region defined by another function g(x,y). That is g(x,y) = 0 is the equation of the boundary. You can just think of this as a curve like the border of a putting green in golf and think of the function f(x,y) as a surface above the region (e.g. the putting green) enclosed by the boundary. You might visualize this surface as a tent or canopy above the putting green. The integral of f(x,y) over this region is analogous to the volume under the canopy and within the putting green.

For large numbers of dimensions this can't be visualized but you can think of an n-dimensional definite integral as an n-dimensional volume (sometimes referred to as "content" rather than "volume" for more than three dimensions).

These multiple integrals come up a lot in problems in engineering, physics and statistics.
(1) If you know the flow velocity as a function of position across the cross-section of a pipe, to get the total flow through the pipe (e.g. in gallons per minute) you would integrate that over the two-dimensional cross-section of the pipe.
(2) In probability and statistics (which are used in all sciences and engineering disciplines as well as other places) you often have some probability model over many variables and you want to compute the total probability for some region of the corresponding space. That region corresponds to a set of possible outcomes. To get that total probability you integrate the probability model (probability density) over that region.

See the Wikipedia page on multiple integrals. I put the link to it under "Sources" below. It has an excellent discussion and lots of example applications.

Answer 2

You're referring to double and triple integrals (more can be done). If you want the area under a single variable function f(x), you can integrate it. If you want the volume under a 2-variable function f(x,y), you can integrate it twice. If you want to find the net sum of all sources and sinks in a vector field within a closed surface, you use the triple integral. And so on. The more degrees of freedom your particular system involves, the more integrals in sequence you might have to use to find out some values. For example, to find the volume of a hypersphere of n dimensions, you need to do n integrals on it. Check link on hyperspheres for this.

Answer 3

in fixing a differential equation, the cf and specific essential are would desire to. with out looking them you cant get an answer. whether specific essential cant be desperate while there's no consistent. if there's a relentless then we would desire to be careful in looking PI.