Question

On a sketch of y=ln(x)y=ln(x), represent the left riemann sum with n=2n=2 approximating ∫65ln(x)dx∫56ln(x)dx. write out the terms of the sum, but do not evaluate it:

Answer 1

Answer: y = f(x) = ln(x) sketch diagram , with 2 rectangles from the left side Sum = area of rectangle 1 + area of rectangle 2 = (1/2)(f(3)+f(3.5)) = ...plug values and calculator 2/ with 2 rectangles from the right side Sum = area of rectangle 1 + area of rectangle 2 = ...

Answer 2

`\text{Left Riemann sum }= (1)/(2)(\ln 5+\ln 5.5)`

Explanation:

We are given y=ln(x)

We need to represent the left riemann sum with n=2

Please see the attachment for sketch and two left rectangle.

`I=\int_5^6 \ln xdx`

Left Riemann sum of integral

`\int_a^bf(x)dx=(b-a)/(n)(f(x_0)+f(x_1))`

where, f(x)=ln(x), a=5 , b=6, n=2 ,
`x_0=5` and
`x_1=5.5`

Now we write given integral into riemann sum

`L_2=(6-5)/(2)(f(5)+f(5.5))`

`L_2=(1)/(2)(\ln 5+\ln 5.5)\approx 1.6571`