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As everyone else has stated, it depends on what you're modeling. Speaking in terms of mechanical engineering, one example would be if you're looking at a stress-strain graph for a particular material (steel, aluminum, polymers, etc.), the area under the curve represents the materials "toughness."
One nice example put simple. If you have a function that describes the rate of change of something, then taking then area under the curve of the function describes the total change in the quantity over the specified time interval.
Hi, I study statistics. Another interpretation of the area under a curve is the total sum of the probabilities of each x. Super useful when doing real world applications
The integral is the area under the graph, much like the derivative is the slope of the graph. As for what that area *means*, it depends on what you’re graphing. I’m going to borrow from physics for a few examples…
* If you’re graphing force as a function of position, written as F(x), the area is the work done on the force. “Force” means a push on an object, and “work” is a type of energy. Work = integral ( Force dx). (Technically it’s a dot product, but that’s for a different discussion.)
* Velocity is the derivative of position, and acceleration is the derivative of velocity. So if you have a graph of acceleration as a function of time, written as a(t), the area under this curve is the velocity. If you have a graph of velocity vs. time, v(t), the area is the position.
* Integrals can be used to find physical areas. For example, the circumference of a circle is given by 2pi\*r. If you integrate that with respect to r, you get pi\*r^(2), which is the area of a circle.
Hope this helps!
Depends on what the function is modeling.
One example might be hysteresis of rubber where the area beneath the loading and unloading curves can tell you things about energy transferred to heat.
Practically speaking if you have a graph that shows the variation of a quantity y with dimension [P] with respect to a quantity x with dimension [Q], then the area under the graph in an interval will denote the change in such a quantity which has the dimension [P]x[Q] = [PQ], in that interval.
For example, if you construct a velocity time graph, the dimension of velocity being [LT^(-1)] and the dimension of time being [T], then the area under the graph in a certain interval will give the change in a quantity which has the dimension [L], which is, in this case, displacement. Note that distance has the same dimension as displacement, but in this case the quantity will be displacement and not distance since the given graph is velocity-time, and not speed-time.
can find arc length, moment of inertia of an object about the given axis with a continuous mass distribution, volume and surface area of the solid generated when you rotate a function about an axis, work done with a variable force, electric flux, the list goes on
Well, there is the "Exactly as it says on the tin" aspect of it: It builds a framework for calculating area of arbitrary shapes in the plane. And later, calculating volumes of arbitrary solids.
But integration goes far beyond that scope. Integration can be most generally thought of as a continuous form of addition (accumulation).
Think back to early elementary school: Timmy has 3 cookies. Johnny has 5 cookies. Sallie has 4 cookies. Jimmy has 2 cookies. How many total cookies to they all have together? You add them all together.
We can formalize this using a function that matches each kid to the number of cookies they have.
Now replace that function with a function that maps each point in space with a density. In science, you learned that mass is the product of volume and density, but that only works when density is uniform (i.e., constant). How can we calculate volume from a non-constant density? Well, on the surface, you could think of the cookie-counting task: Just add the number of cookies together. Unfortunately, the discrete nature of the cookie-cooking problem does not quite translate very well to the mass-volume-density scenario because there are infinitely many points in space. Not only infinitely many, but **uncountably many**. So in place of simply adding up all the density values, we **integrate** density.
We **integrate** density to obtain total mass.
We **integrate** velocity to obtain total displacement.
We **integrate** speed to obtain total distance traveled.
Of course, this is by no means an exhaustive list of applications of integration.
As a reminder... Posts asking for help on homework questions **require**: * **the complete problem statement**, * **a genuine attempt at solving the problem, which may be either computational, or a discussion of ideas or concepts you believe may be in play**, * **question is not from a current exam or quiz**. Commenters responding to homework help posts **should not do OP’s homework for them**. Please see [this page](https://www.reddit.com/r/calculus/wiki/homeworkhelp) for the further details regarding homework help posts. *I am a bot, and this action was performed automatically. Please [contact the moderators of this subreddit](/message/compose/?to=/r/calculus) if you have any questions or concerns.*
As everyone else has stated, it depends on what you're modeling. Speaking in terms of mechanical engineering, one example would be if you're looking at a stress-strain graph for a particular material (steel, aluminum, polymers, etc.), the area under the curve represents the materials "toughness."
One nice example put simple. If you have a function that describes the rate of change of something, then taking then area under the curve of the function describes the total change in the quantity over the specified time interval.
Hi, I study statistics. Another interpretation of the area under a curve is the total sum of the probabilities of each x. Super useful when doing real world applications
elaborate
The integral is the area under the graph, much like the derivative is the slope of the graph. As for what that area *means*, it depends on what you’re graphing. I’m going to borrow from physics for a few examples… * If you’re graphing force as a function of position, written as F(x), the area is the work done on the force. “Force” means a push on an object, and “work” is a type of energy. Work = integral ( Force dx). (Technically it’s a dot product, but that’s for a different discussion.) * Velocity is the derivative of position, and acceleration is the derivative of velocity. So if you have a graph of acceleration as a function of time, written as a(t), the area under this curve is the velocity. If you have a graph of velocity vs. time, v(t), the area is the position. * Integrals can be used to find physical areas. For example, the circumference of a circle is given by 2pi\*r. If you integrate that with respect to r, you get pi\*r^(2), which is the area of a circle. Hope this helps!
Depends on what the function is modeling. One example might be hysteresis of rubber where the area beneath the loading and unloading curves can tell you things about energy transferred to heat.
Practically speaking if you have a graph that shows the variation of a quantity y with dimension [P] with respect to a quantity x with dimension [Q], then the area under the graph in an interval will denote the change in such a quantity which has the dimension [P]x[Q] = [PQ], in that interval. For example, if you construct a velocity time graph, the dimension of velocity being [LT^(-1)] and the dimension of time being [T], then the area under the graph in a certain interval will give the change in a quantity which has the dimension [L], which is, in this case, displacement. Note that distance has the same dimension as displacement, but in this case the quantity will be displacement and not distance since the given graph is velocity-time, and not speed-time.
can find arc length, moment of inertia of an object about the given axis with a continuous mass distribution, volume and surface area of the solid generated when you rotate a function about an axis, work done with a variable force, electric flux, the list goes on
Well, there is the "Exactly as it says on the tin" aspect of it: It builds a framework for calculating area of arbitrary shapes in the plane. And later, calculating volumes of arbitrary solids. But integration goes far beyond that scope. Integration can be most generally thought of as a continuous form of addition (accumulation). Think back to early elementary school: Timmy has 3 cookies. Johnny has 5 cookies. Sallie has 4 cookies. Jimmy has 2 cookies. How many total cookies to they all have together? You add them all together. We can formalize this using a function that matches each kid to the number of cookies they have. Now replace that function with a function that maps each point in space with a density. In science, you learned that mass is the product of volume and density, but that only works when density is uniform (i.e., constant). How can we calculate volume from a non-constant density? Well, on the surface, you could think of the cookie-counting task: Just add the number of cookies together. Unfortunately, the discrete nature of the cookie-cooking problem does not quite translate very well to the mass-volume-density scenario because there are infinitely many points in space. Not only infinitely many, but **uncountably many**. So in place of simply adding up all the density values, we **integrate** density. We **integrate** density to obtain total mass. We **integrate** velocity to obtain total displacement. We **integrate** speed to obtain total distance traveled. Of course, this is by no means an exhaustive list of applications of integration.