How Does Using Midpoints in Integrals Estimate Area?

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In summary, midpoints in integrals are a method used to approximate the area under a curve by dividing it into smaller rectangles and finding the average height of each rectangle. The midpoint of each subinterval is used as the height of the rectangle, and the width of the subinterval is used as the width of the rectangle. The purpose of using midpoints in integrals is to approximate the area under a curve when the exact value cannot be calculated. However, it can only provide an approximate value and becomes less accurate with sharp turns or large subinterval widths.
  • #1
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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  • #2
What part of the solution did you not understand?
 

1. What are midpoints in integrals?

Midpoints in integrals are a method used to approximate the area under a curve by dividing it into smaller rectangles and finding the average height of each rectangle. This average height is then multiplied by the width of the rectangle to find the approximate area.

2. How are midpoints calculated in integrals?

To calculate midpoints in integrals, you first need to determine the interval over which you want to find the area. Then, divide this interval into smaller subintervals of equal width. The midpoint of each subinterval is then used as the height of the rectangle, and the width of the subinterval is used as the width of the rectangle.

3. What is the purpose of using midpoints in integrals?

The purpose of using midpoints in integrals is to approximate the area under a curve when the exact value cannot be calculated. This method is often used in situations where the function is not easily integrable or when numerical methods are needed.

4. Can midpoints in integrals provide an exact value for the area under a curve?

No, midpoints in integrals can only provide an approximate value for the area under a curve. This method relies on dividing the curve into smaller rectangles, which can never perfectly match the shape of the curve. However, as the number of subintervals increases, the approximation becomes more accurate.

5. What are some limitations of using midpoints in integrals?

One of the limitations of using midpoints in integrals is that it can only provide an approximation and not an exact value. This method also becomes less accurate when the curve has sharp turns or when the width of the subintervals is too large. Additionally, it can be time-consuming to calculate midpoints for a large number of subintervals.

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