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Trapezoidal rule

# Trapezoidal sums

AP.CALC:
LIM‑5 (EU)
,
LIM‑5.A (LO)
,
LIM‑5.A.1 (EK)
,
LIM‑5.A.2 (EK)
,
LIM‑5.A.3 (EK)
,
LIM‑5.A.4 (EK)
The area under a curve is commonly approximated using rectangles (e.g. left, right, and midpoint Riemann sums), but it can also be approximated by trapezoids. Trapezoidal sums actually give a better approximation, in general, than rectangular sums that use the same number of subdivisions. Created by Sal Khan.

## Want to join the conversation?

• can you show how to find the exact area using limit approaching infinite?
• You would take the same fundamental concept, except apply it to a summation. As in the Limitation as N->Infinity, being apply the formula to the following:

1/2[f(x0) + 2f(x1) + 2f(x3) ... 2f(xn)]

You would apply this concept to a summation series:

Σ 1/2[f(n)+ 2f(n+1) + 2f(n+2) + 2f(n+3)... f(n+n)]
n=0

Simple from here, you would evaluate using the function and determine if the sum diverges or converges. If it converges, that will be your area, if it diverges, well it then diverges.
• I feel like using middle boudaries for functions that are only concave upwards or only concave downwards do much better approximation. Is that right?
• Yes, you are right, middle boundaries are more accurate for functions of 1 concavity, but the trapezoidal method works a lot better for functions with many concavities.
• Does the Simpson rule has a similar geometrical explanation or is it a different concept?
• The trapezoid rule uses the average between points to approximate the line the graph makes between the two points.
Simpson's rule uses a quadratic parabolic arc.
Simpson's is usually more accurate and quicker computationally than the trapezoid rule since it converges faster - that is, it gives a better result with fewer subdivisions because it "hugs the curve" better.
• So is Simpson's Rule essentially the same process as this, in terms of the necessary solving equation?
• The processes are similar. The trapezoid rule joins f(n) and f(n+1) with a straight line (that is, it just uses 2 points) while Simpson's uses 3 points, f(n), f(n+1) PLUS a midpoint. These three points are used to describe a parabola, which is a closer approximation to the curve f than just the straight line approximation that the trapezoid rule gives.
Can a more accurate approximation for the area under the curve be reached by calculating the slope of the function at each midpoint [i.e. Fprime(Xsub1+Xsub2 / 2)] and using that line to define the heights of each trapezoid? Is it only more accurate if one uses sufficiently small (infinitesimal?) increments of delta X?
-Or-
Is that essentially what Sal is already doing at ?
• What you're asking is not essentially the same as Sal does @, but I do think it would yield more accurate results. I wouldn't know for sure and it might be more accurate for some functions and less for others. Try to find it out yourself, it is an interesting thought! Maybe other websites have info on Riemann sums with slopes of the midpoints.
• How do you calculate Riemann sums with unequal widths?
• You would just find the area of each section and then add them together. Since they are not equal widths, you have to find each area separately.
• Do we have to split the curve into 5 trapizoids?
• ^To follow up with The Last Guy, like the RAM method, the greater the N value is the more accurate it is. An infinity amount of trapezoids with a minuscule base would make it more accurate.
• This is just a random question, but wouldn't the exact area of the space under x^2 from -3 to 3 be 18? It's just this is a rather round number for the area under a curve.
Note:
I did chose the constant C to be 0 just for simplicity.
• The only reason that works is that you chose your limits to be -3 and 3. The antiderivative of x^2 is (x^3)/3 and when you are evaluating x=3, then the three in the denominator acts to reduce the power of the expression by 1 so you get back to x^2. If you try any other number as your limits of integration you will not get the same result as 2•x^2.

Also you only use the constant C for the indefinite integral of a function. When you are evaluating the definite integral you are evaluating the antiderivative of the integrand at the upper limit and subtracting by the antiderivative of the integrand evaluated at the lower limit. Since you are evaluating the same function, just at different points, you will have the same constant and subtracting it from the other results in Zero anyway. By convention when you show the result of integration of a definite integral you show your function without the + C, but you put a bracket ] after the function and write your limits of integration after it, indicating that you will be evaluating the function in this range.
• what is importance of knowing the area under the curve?
• Finding the area under the curve is just the first, most intuitive way to think about the process of integration, which is where these Riemann sums are leading. Another immediate use of area under the curve, acronymed as AUC by pharmacologists, plots the concentration of a drug in the blood plasma over time - that is, the AUC represents the total drug exposure over time.

Later, when you get into double and triple integrals, you can calculate volumes, centers of mass, moments of inertia, vector field divergence and curl. They also have extensive use in probability theory, especially when you get into quantum mechanics to describe probability distributions of two or more random variables. Finally, but not exhaustively, integration provides a way to convolve functions. Convolution essentially shows how the properties "shape" one function are modified by another function.

I take acoustical impulse responses in an area, like the inside of a car, or from behind a door, and convolve them with a spoken voice or other sound to make it sound like the sounds were actually recorded in the area where the impulse response was taken. This is a very common technique in film and music post-production, in which I work, from time to time.

In the end, area under the curve is just one way of thinking about what the process of integration represents, but you will find there are many ways of interpreting the results of integration, depending on the function(s) being integrated.