# How do you use the midpoint rule to estimate area?

show below please.

A good way to approximate areas with rectangles is to make each rectangle cross the curve at the midpoint of that rectangles top side.

A midpoint sum is a much better estimate of area than either a left-rectangle or right-rectangle sum.

The figure below shows you why it is better:

You can see in the figure that the part of each rectangle that’s above the curve looks about the same size as the gap between the rectangle and the curve.

A midpoint sum produces such a good estimate because these two errors roughly cancel out each other.

You can approximate the exact area under a curve between a and b

with a sum of midpoint rectangles given by the following formula.

In general, the more rectangles, the better the estimate:

Where, n is the number of rectangles

is the width of each rectangle, and the function values are the heights of the rectangles.

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To use the midpoint rule to estimate the area under a curve, follow these steps:

- Divide the interval ([a, b]) into (n) subintervals of equal width. The width of each subinterval is ( \Delta x = \frac{b - a}{n} ).
- Find the midpoint of each subinterval. The midpoint of the (i)th subinterval is ( x_i = a + \frac{(2i - 1)\Delta x}{2} ).
- Evaluate the function at each midpoint to find the corresponding function values ( f(x_i) ).
- Multiply each function value by the width of the subinterval to find the area of the rectangle formed by the midpoint and the function value.
- Sum up all the areas of the rectangles to estimate the total area under the curve.

The formula for estimating the area using the midpoint rule is:

[ \text{Area} \approx \sum_{i=1}^{n} f(x_i) \Delta x ]

Where ( f(x_i) ) is the value of the function at the midpoint ( x_i ) of the (i)th subinterval, and ( \Delta x ) is the width of each subinterval.

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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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