The integral of a function fx between the points a and b is denoted by
and can be roughly described as the area below the graph of y=fx and above the x-axis, minus any area above the graph and below the x-axis, and all taken between the points a and b.
The integral is important because it is an antiderivative for the original function, that is, if
A Riemann sum is an approximation to the integral, that is, an approximation using rectangles to the area mentioned above. The line segment from x=a to x=b is split into n subsegments which form the bases of these rectangles, and the corresponding heights are determined by the value of fxi at some point xi between the endpoints of the subsegment. The division of the segment a,b into subsegments is called a partition. For the sake of convenience we will assume here that the subsegments are of equal width, although this is not strictly necessary.
The Riemann Sum is then given by the general formula:
∑i=1n fxi ⋅b−an
There are five main types of Riemann Sums, depending on which point xi is chosen to determine the height:
Right Sum: the right endpoint of the subsegment
Left Sum: the left endpoint of the subsegment
Middle Sum: the point half way between the left and right endpoints
Lower Sum: any point xi such that fxi is minimal
Upper Sum: any point xi such that fxi is maximal
Instead of using rectangles to approximate the area under the curve, trapezoids give a better approximation to the area.
The area of a trapezoid with base b and heights p and q is given by:
Area = b2p+q
To approximate the area under the curve, add up the area of all the trapezoids.
Area =12y0 + y1Δx + 12y1 + y2Δx + 12y2 + y3Δx ...
which can be simplified to become the trapezoidal rule:
Area ≈ Δxy02 + y1+ y2 + y3+..+yn2
Adjust the number of trapezoids used to approximate the area under the curve.
Example = Example1Example2Example3Example4
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