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How do you use the Midpoint Rule with #n=5# to approximate the integral #int_1^(2)1/xdx# ?

Answer 1

Cameron Alexander

The interval #[1,2]# is divided into 5 equal subintervals

#[1,1.2],[1.2,1.4],[1.4,1.6],[1.6,1.8], and [1.8,2]#.

Each interval are of length #Delta x={b-a}/n={2-1}/5=0.2#.

The midpoints of the above subintervals are

#1.1,1.3,1.5,1.7, and 1.9#.

Using the above midpoints to determine the heights of the approximating rectangles, we have

#M_5=[f(1.1)+f(1.3)+f(1.5)+f(1.7)+f(1.9)]Delta x#

#=(1/1.1+1/1.3+1/1.5+1/1.7+1/1.9)cdot 0.2 approx 0.692#

By Midpoint Rule,

#int_1^2 1/x dx approx 0.692#.

I hope that this was helpful.

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Answer 2

Axel Alvarez

To approximate the integral ( \int_1^2 \frac{1}{x} , dx ) using the Midpoint Rule with ( n = 5 ), divide the interval ( [1, 2] ) into ( n = 5 ) subintervals of equal width. Then, find the midpoint of each subinterval and evaluate the function at these midpoints. Finally, multiply each function value by the width of the subintervals and sum them up to get the approximation of the integral.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.