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ryanuser
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We can find an area under a curve with integral, but how if the curve is under the x axis? Do e still use integral?
In general where can we use integral?
In general where can we use integral?
ryanuser said:We can find an area under a curve with integral, but how if the curve is under the x axis? Do e still use integral?
HallsofIvy said:...There is no such thing as negative area (and "area" is the correct word here, not "space"). The integral as area is one simple application. When you integrate
f(x)=x3
from -1 to 1, you are NOT calculating an area. Strictly speaking when you integrate any
∫baf(t)dt
you are NOT calculating an area unless that is the particular application you are using the integral for. The integral of a function, from a to b, say, is a number. What that number means depends on the application.
ryanuser said:In general where can we use integral?
I don't mean to be rude, but you need dx in the integral for that to be true. You could confuse some people, otherwise.EJChoi said:If the curve is under the x-axis the integral has a negative value.
Basically this gives you the net area under the curve.
If you want to find the total area both under and above the x-axis, you'll need to integrate the modulus of the function.
i.e [tex] \displaystyle \int^{2\pi}_{0} sin(x) = 0[/tex]
But..
[tex] \displaystyle \int^{2\pi}_{0} |sin(x)| = 4 [/tex]
Goa'uld said:I don't mean to be rude, but you need dx in the integral for that to be true. You could confuse some people, otherwise.
ryanuser said:...
In general where can we use integral?
This is only a sufficient condition; it is not a necessary condition. The condition of Riemann integrable is much more general. The indicator function associated with the Cantor set has uncountably many discontinuities but it is Riemann integrable.dkotschessaa said:If a function is continuous on an interval, it will be integrable on that interval. (That's a mouthful!)
dkotschessaa said:If a function is continuous on an interval, it will be integrable on that interval. (That's a mouthful!)
Indeed, we can go a bit further and state a necessary and sufficient condition for Riemann integrability of a bounded function ##f## defined on a closed, bounded interval: ##f## is Riemann integrable if and only if the set of points where it is discontinuous has Lebesgue measure zero. The indicator function of the Cantor set qualifies because its set of discontinuities is precisely the Cantor set, which has measure zero.WannabeNewton said:This is only a sufficient condition; it is not a necessary condition. The condition of Riemann integrable is much more general. The indicator function associated with the Cantor set has uncountably many discontinuities but it is Riemann integrable.
An integral is a mathematical concept that represents the area under a curve in a graph. It is used to calculate the total value of a quantity that is changing over a certain range.
The purpose of an integral is to find the total value of a quantity that is changing continuously. It is often used in real-world applications to solve problems related to motion, volume, and area.
A definite integral has specific limits of integration, meaning it calculates the area under a curve between two specific points. An indefinite integral has no limits of integration and is used to find the general antiderivative of a function.
An integral is calculated using a process called integration, which involves breaking down the area under a curve into smaller, simpler shapes and summing them up to find the total area. This process can be done analytically or numerically using various techniques.
Integrals are used in various fields such as physics, engineering, economics, and statistics. Some common real-world applications include calculating the distance traveled by an object, finding the volume of a 3D shape, and determining the total cost of a continuously changing variable.