How do you use Riemann sums to evaluate the area under the curve of #f(x)= In(x)# on the closed interval [3,18], with n=3 rectangles using right, left, and midpoints?
Please see the explanation section below.
I will use what I think is the usual notation throughout this solution.
The subintervals then are:
So, using left endpoints, we have
The arithmetic is left to the student.
Using right endpoints we have
The arithmetic is left to the student.
Using midpoints we have
The arithmetic is left to the student.
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To evaluate the area under the curve of ( f(x) = \ln(x) ) on the closed interval ([3,18]) using Riemann sums with ( n = 3 ) rectangles, you can use right, left, and midpoint methods.

Right Riemann Sum: [ \text{Right Riemann Sum} = \sum_{i=1}^{n} f(x_i) \Delta x ] [ = f(x_1) \Delta x + f(x_2) \Delta x + f(x_3) \Delta x ] [ = \ln(x_1) \Delta x + \ln(x_2) \Delta x + \ln(x_3) \Delta x ] [ = \ln(6) \cdot 3 + \ln(9) \cdot 3 + \ln(12) \cdot 3 ]

Left Riemann Sum: [ \text{Left Riemann Sum} = \sum_{i=1}^{n} f(x_{i1}) \Delta x ] [ = f(x_0) \Delta x + f(x_1) \Delta x + f(x_2) \Delta x ] [ = \ln(x_0) \Delta x + \ln(x_1) \Delta x + \ln(x_2) \Delta x ] [ = \ln(3) \cdot 3 + \ln(6) \cdot 3 + \ln(9) \cdot 3 ]

Midpoint Riemann Sum: [ \text{Midpoint Riemann Sum} = \sum_{i=1}^{n} f\left(\frac{x_{i1} + x_i}{2}\right) \Delta x ] [ = f\left(\frac{x_0 + x_1}{2}\right) \Delta x + f\left(\frac{x_1 + x_2}{2}\right) \Delta x + f\left(\frac{x_2 + x_3}{2}\right) \Delta x ] [ = \ln\left(\frac{3 + 6}{2}\right) \cdot 3 + \ln\left(\frac{6 + 9}{2}\right) \cdot 3 + \ln\left(\frac{9 + 12}{2}\right) \cdot 3 ]
After calculating the values, you can sum them up to find the approximate area under the curve using each method.
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When evaluating a onesided 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.
When evaluating a onesided 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.
When evaluating a onesided 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.
When evaluating a onesided 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.
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