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What are the different ways of finding the area under a curve and how do they work? (Calculus) dfsuhduhfidlafjglkjdsg

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Sorry I didn't see this until today.If I am understanding what you are asking my answer would be:
In calculus you find the area under a curve by taking the integral.
To fine the area between two specific places you evaluate the definite integral between those limits.
There are numerical ways to find the area or to evaluate an integral.
You split the region under the curve between the two limits into finite slices (columns).
There are many different ways you can do this.
Two common ways to do this are with a rectangular slice or a trapezoidal slice.
For the rectangular method you can have the top left of the rectangle be a point on the curve, have the top right be a point on the curve, or the more accurate way is to have the center point of the top of the rectangle by on the curve.  That way there will be a little of the rectangle above the curve and a little below coming to an average of about right.  Then you calculate the area of all these little tall rectangles and add them up.  On a computer this is easy and you can have the computer keep making the widths of the rectangles smaller and smaller -> approaching a limit of zero or some value of precision you want.
Sorry we cannot put diagrams here so it is very hard to explain.
Look up Numerical Integration.
This place has a diagram to see what I mean:
For the trapezoidal method every thing is essentially the same.
You make small slices of the area under the curve.
The difference here is you make a trapezoid so the top edge of the slice is not flat but is sloped (i.e. you have a trapezoid).  Then you calculate the areas of the narrow tall trapezoids and add them up.  Because the top is sloped along the curve it gives a better approximation of the area than a rectangle.  On a calculator, again, this is easy to do all these small calculations and add them.  You can also make the trapezoids get narrower and approach zero to get better results.
Look up Trapezoidal rule / numerical integration
Here is a page that has a diagram of both the rectangles and trapezoids:
Compare the two wiki pages,  also there are things on Kahn Academy and on UTube.