How Does Using Midpoints in Integrals Estimate Area?

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SUMMARY

The discussion focuses on estimating the area under the curve of the function f(x) = e^-x from x=0 to x=2 using the midpoint rule with four sub-intervals. The calculated area, A, is approximately 0.8357. The midpoint values for the sub-intervals are derived as x*1=0.25, x*2=1.25, x*3=1.75, and x*4=2.25. The user expresses confusion regarding the solution process, indicating a need for clarification on specific steps.

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  • Basic knowledge of the exponential function f(x) = e^-x
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moaath
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1. let A be the area of the region that lies
under the graph of f(x) = e^-x , between x=0 and x=2.

1-Estimate the area using the sample point to be the midpoint and using fuor sub-intervals.
x*1=(0+0.5)\2=1\4.
x*2=(1+0.5)\2=3\4.
x*3=(1+3\2)\2=5\4.
x*4=(2+3\2)\2=7\4
A=segma i=1 to n f(x*i) X delta x =0.8357

I copied the answer from the board with my doctor but I didn't understand the solution
 
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