Using the Midpoint Rule In Exercises 15-20, use the Midpoint Rule with n = 4 to approximate the area of the region bounded by the graph of f and the x-axis over the interval. Sketch the region. See Examples 2 and 3.
To calculate: The area bounded by the function in the interval by mid-point rule. Also plot the graph between function f and x-axis.
The provided function is . area of this function is calculated on the interval and the provided interval will be divided into 4 sub interval as .
The approximate area of any definite integral by the use of midpoint rule is calculated by the use of following steps.
Step1: firstly, divide the provided interval of the function into n sub intervals by the use of formula;
where are the intervals and n is the subinterval value and is the width.
Step2: by the use of above calculated value of subinterval find the midpoint of each sub interval.
Step3: final step is to obtain approximate area by calculating function f at each mid-point from the use of formula;
Consider the provided function;
Now, put n equal to 4, and in the above mid-point rule as;
Width of the provided interval will be .
So, the interval with width will be divided into 4 subintervals as;
Now, the value of midpoint of each sub interval is calculated as;
where are the upper value and lower value of the subintervals
So, for subinterval mid-point is ;
Similarly for subinterval mid-point is ;
For subinterval mid-point is ;
So, the mid points of sub intervals are respectively
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