An area is approximated to be 14 in 2 using a left-endpoint rectangle approximation method. a right- endpoint approximation of the same area is equal to 21 in 2. assuming the same number of equal subdivision for all approximation methods, what would the result of a trapezoid approximation for the area?

The trapezoidal approximation will be the average of the left- and right-endpoint approximations.

Let's consider a simple example of estimating the value of a general definite integral,



Split up the interval into equal subintervals,



where and . Each subinterval has measure (width) .

Now denote the left- and right-endpoint approximations by and , respectively. The left-endpoint approximation consists of rectangles whose heights are determined by the left-endpoints of each subinterval. These are . Meanwhile, the right-endpoint approximation involves rectangles with heights determined by the right endpoints, .

So, you have




Now let denote the trapezoidal approximation. The area of each trapezoidal subdivision is given by the product of each subinterval's width and the average of the heights given by the endpoints of each subinterval. That is,



Factoring out and regrouping the terms, you have



which is equivalent to



and is the average of and .

So the trapezoidal approximation for your problem should be

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