Table of Contents

## How do you find the midpoint of the area under a curve?

Use the midpoint rule to approximate the area under a curve given by the function f ( x ) = x 2 + 5 f(x)=x^2+5 f(x)=x2+5 on the interval [0,4] and n=4. Now we will find the values halfway across, this will give us the midpoint of the sub-intervals.

### What is the midpoint rule Riemann sum?

In a midpoint Riemann sum, the height of each rectangle is equal to the value of the function at the midpoint of its base. We can also use trapezoids to approximate the area (this is called trapezoidal rule). In this case, each trapezoid touches the curve at both of its top vertices.

#### What is the midpoint rule in integrals?

1: The midpoint rule approximates the area between the graph of f(x) and the x-axis by summing the areas of rectangles with midpoints that are points on f(x).

**Why is Simpson’s rule more accurate?**

The reason Simpson’s rule is more accurate is that it’s matching a parabola to the curve, rather than a straight line. Simpson’s rule gives the exact area beneath the graphs of functions of degree two or less (parabolas and straight lines), while the other methods are only exact for functions whose graphs are linear.

**How do you find the area of a midpoint of a rectangle?**

A midpoint sum produces such a good estimate because these two errors roughly cancel out each other. from 0 to 3. For the three rectangles, their widths are 1 and their heights are f(0.5) = 1.25, f(1.5) = 3.25, and f(2.5) = 7.25. Area = base x height, so add 1.25 + 3.25 + 7.25 to get the total area of 11.75.

## Is a midpoint Riemann sum an over or underestimate?

(To see why, draw a sketch.) If the graph is concave up the trapezoid approximation is an overestimate and the midpoint is an underestimate. If the graph is concave down then trapezoids give an underestimate and the midpoint an overestimate.

### How do you find a midpoint?

Example: Find the Midpoint

- First, add the x coordinates and divide by 2. This gives you the x-coordinate of the midpoint, xM
- Second, add the y coordinates and divide by 2. This gives you the y-coordinate of the midpoint, yM
- Take each result to get the midpoint. In this example the midpoint is (9, 5).

#### Is the midpoint rule always more accurate than the trapezoidal rule?

The Midpoint rule is always more accurate than the Trapezoid rule. For example, make a function which is linear except it has nar- row spikes at the midpoints of the subdivided intervals. Then the approx- imating rectangles for the midpoint rule will rise up to the level of the spikes, and be a huge overestimate.

**Which one is better trapezoidal or Simpsons?**

14.2. Simpson’s rule is a method of numerical integration which is a good deal more accurate than the Trapezoidal rule, and should always be used before you try anything fancier.

**Is Simpson or trapezoidal better?**

In trapezoidal we take every interval as it is . In simpson’s we further divide it into 2 parts and then apply the formula. Hence Simpson’s is more precise.

## How do you find the area under the left endpoint of a curve?

Figure 5.1. 7: With a left-endpoint approximation and dividing the region from a to b into four equal intervals, the area under the curve is approximately equal to the sum of the areas of the rectangles.