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Chapter 4: Q. 57 (page 400)

For the logarithmic value, use Riemann sums with at least four rectangles to find an over approximation and an under-approximation for the value.

lne

Short Answer

Expert verified

Using left and right sums with four rectangles is 0.8999鈮�lne鈮�1.149.

Step by step solution

01

Step 1. Given information.

Consider the given question,

lne

02

Step 2. Use Riemann sums for left sum definition.

Consider the function,

fx=1t

The left-sum defined for n rectangles on a,bis 鈭�k=1nfxk-1鈻�x.

Where, 鈻�x=b-an,xk=a+k鈻�x

Now 鈻�x,

=2.8-014=28140=0.2

03

Step 3. Find the left sum.

Consider the given question,

xk=0+k0.2=0.2k

In the left sum, xk-1is the leftmost point in the interval xk-1,xk.

So, xk-1=k-10.2=0.2k-0.2

The left sum is given below,

鈭�k=02.8ln0.2k-0.20.2鈮�0.8999

04

Step 5. Find the right sum.

The right-sum defined for n rectangles on a,bis 鈭�k=1nfx鈻�x.

The right sum is given below,

=鈭�k=02.8ln0.2k-0.20.2鈮�1.149

Therefore, the value oflneis0.8999鈮�lne鈮�1.149.

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