How do you use the midpoint rule to estimate area?

Question

Isaiah Allen · Accepted Answer

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.

#"Midpoint Rectangle Rule"#

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

#A=int_a^by*dx#

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

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

#A_"Midpoint"=(b-a)/n[f((x_0+x_1)/2)+f((x_1+x_2)/2)+...+f((x_n-1+x_n)/2)]#

Where, n is the number of rectangles

#width=(b-a)/2#

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

Ezra Adams · Answer

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

1. Divide the interval $[a, b]$ into $n$ subintervals of equal width. The width of each subinterval is $ \Delta x = \frac{b - a}{n} $.
2. Find the midpoint of each subinterval. The midpoint of the $i$th subinterval is $ x_i = a + \frac{(2i - 1)\Delta x}{2} $.
3. Evaluate the function at each midpoint to find the corresponding function values $ f(x_i) $.
4. 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.
5. 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.