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Chapter 4: Q. 55 (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.

ln10

Short Answer

Expert verified

Using left and right sums with 9rectangles is1.928鈮�ln10鈮�2.8289.

Step by step solution

01

Step 1. Given information.

Consider the given question,

ln10

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,

=10-19=99=1

03

Step 3. Find the left sum.

Consider the given question,

xk=1+k1=k+1

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

So, xk-1=k

The left sum is given below,

=鈭�k=1101k1=1+12+13+14+15+16+17+18+19+110鈮�2.828897

04

Step 4. Use Riemann sums for right sum definition.

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

Where, 鈻�x=b-an,xk=a+k鈻�x.

Now 鈻�x,

=10-19=99=1

05

Step 5. Find the right sum.

Consider the given question,

xk=1+k1=k+1

The right sum is given below,

=鈭�k=1101k+11=12+13+14+15+16+17+18+19+110+111鈮�1.9287

Therefore, the value ofln10is1.928鈮�ln10鈮�2.8289.

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