# How do you use Riemann sums to evaluate the area under the curve of #1/x# on the closed interval [0,2], with n=4 rectangles using midpoint?

I will use what I think is the usual notation throughout this solution.

The subintervals are:

The midpoints are:

Now the Riemann sum is the sum of the area of the 4 rectangles. We find the area of each rectangle by

Here we are using midpoints for sample points. So

Finish the arithmetic to finish.

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To use Riemann sums to evaluate the area under the curve of (1/x) on the closed interval ([0, 2]) with (n = 4) rectangles using the midpoint method, follow these steps:

- Divide the interval ([0, 2]) into (n) subintervals of equal width. Since (n = 4), each subinterval will have a width of (2/4 = 0.5).
- Determine the midpoints of each subinterval. For (n = 4), the midpoints will be at (x = 0.25), (x = 0.75), (x = 1.25), and (x = 1.75).
- Evaluate the function (1/x) at each midpoint to get the corresponding function values.
- Multiply each function value by the width of the subinterval ((0.5) in this case)).
- Sum up all these products to get the Riemann sum.

Mathematically, the Riemann sum (S) for (n) rectangles using the midpoint method is given by:

[ S = \sum_{i=1}^{n} f\left(\frac{x_{i-1} + x_i}{2}\right) \cdot \Delta x ]

Where:

- (f(x)) is the function (1/x).
- (x_{i-1}) and (x_i) are the endpoints of the (i)th subinterval.
- (\Delta x) is the width of each subinterval.

Substituting the values for the midpoint method with (n = 4) and evaluating at the midpoints:

[ S = \left( f(0.125) + f(0.375) + f(0.625) + f(0.875) \right) \cdot 0.5 ]

Then, calculate the function values at the midpoints and perform the summation to find the Riemann sum.

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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.

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.

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.

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