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.

Function

Interval

f ( x ) = x 3 + 6               [ 1 , 5 ]

To calculate: The area bounded by the function f(x)=x3+6 in the interval [1,5] by mid-point rule. Also plot the graph between function f and x-axis.

Given Information:

The provided function is f(x)=x3+6. area of this function is calculated on the interval [1,5] and the provided interval will be divided into 4 sub interval as n=4.

Formula used:

The approximate area of any definite integral ∫baf(x)dx 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;

Δx=b−an where [a,b] are the intervals and n is the subinterval value. and Δx 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;

∫abf(x)dx=b−an[f(x1)+f(x2)+f(x3)+⋯+f(xn)].

Calculation:

Consider the provided function;

Now, put n equal to 4, a=0 and b=1 in the above mid-point rule as;

Δx=5−14Δx=1

Width Δx of the provided interval will be 1.

So, the interval [1,5] with width 1 will be divided into 4 subintervals as;

[1,2], [2,3], [3,4], [4,5].

Now, the value of midpoint of each sub interval is calculated as;

[x1,x2]=x1+x22 where x1,x2 are the upper value and lower value of the subintervals, for subinterval [1,2] mid-point is ;

[1,2]=1+22=32

For subinterval [2,3] mid-point is ;

[2,3]=2+32=52

For subinterval [3,4] mid-point is ;

[3,4]=4+32=72

For subinterval [4,5] mid-point is ;

[4,5]=4+52=92

So, the mid points of sub intervals are 32, 52, 72, 92 respectively