# Using Trapezoid Rule Twice huh?

• issh
In summary, the conversation revolves around using the trapezoidal rule to calculate the area and volume of a pit. The formula A=(1/2)w[E+2M] is used, where w is the width between each interval, E is the sum of the end values, and M is the sum of the middle values. The conversation also discusses using the trapezoidal rule in a 2-dimensional integral to estimate the volume of water in the pit.
issh

## Homework Equations

A=(1/2)w[E+2M]
Where w=width between each interval
E=f(a0)+f(an) is the sum of the end values
M=f(a1)+f(a2)+...f(an-1)

## The Attempt at a Solution

I found the area of the first trapezoidal rule but can't identify the other
w=(15/5)=3metres
E=2.6+2.4=5
M=3+3.2+2.9+2.4=11.5

Therefore, A=(1/2)w[E+2M]=42

I couldn't identify the 2nd one

In order to calculate the volume of water in the pit, you can calculate the area of each of the 6 cross sections using the trapezoidal rule. Make a table or plot of the resulting cross sectional areas and use the trapezoidal rule again in the direction perpendicular to the cross sections to calculate the volume of water in the pit.

You are estimating a volume, so you are doing a 2-dimensional integral of the form [tex] \int \int f(x,y) \, dx \, dy [\tex] where f(x,y) = depth at point (x,y). For each x, you do the trapezoidal rule to estimate $\int f(x,y)\, dy$, and you are doing that at the x-points 0, 3, 6, 9, 12 (meters). Of course, at different x you have different numbers of y-points, so rather than doing the trapezoidal rule twice, I would rather say you are doing it 5 times (once for each of the 5 x-values).

RGV

Shouldn't this be in Calculus and Beyond?

## 1. How does the Trapezoid Rule work?

The Trapezoid Rule is a method for estimating the area under a curve. It divides the area into trapezoids and calculates the sum of their areas to approximate the total area.

## 2. Why would you use the Trapezoid Rule twice?

The Trapezoid Rule can be used twice to increase the accuracy of the estimated area. By dividing the interval into smaller subintervals and applying the Trapezoid Rule to each subinterval, the estimated area becomes more precise.

## 3. What is the formula for the Trapezoid Rule?

The formula for the Trapezoid Rule is A ≈ (b-a)/2 * (f(a) + f(b)), where a and b are the endpoints of the interval and f(a) and f(b) are the function values at those points.

## 4. What are the limitations of using the Trapezoid Rule twice?

The Trapezoid Rule can only provide an estimate of the actual area and is not always accurate. Using it twice can improve the accuracy, but it is still limited by the number of subintervals used and the behavior of the function.

## 5. In what situations would using the Trapezoid Rule twice be beneficial?

Using the Trapezoid Rule twice can be beneficial when the function is not easily integrable or when the estimated area needs to be more precise. It can also be useful when the function has a lot of variation in a small interval, as dividing it into smaller subintervals can help capture that variation.

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