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A graph of the temperature in Boston on August 11, 2013, is shown. Use Simpson's Rule with $ n = 12 $ to estimate the average temperature on that day.

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$\int_{0}^{24} T(t) d t \approx S_{12}$ and $T_{\text {ave }}=\frac{1}{24-0} \int_{0}^{24} T(t) d t \approx 70.84^{\circ} \mathrm{F}$

00:47

Yuou Sun

Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 7

Approximate Integration

Integration Techniques

Campbell University

Harvey Mudd College

Baylor University

Lectures

01:53

In mathematics, integration is one of the two main operations in calculus, with its inverse, differentiation, being the other. Given a function of a real variable, an antiderivative, integral, or integrand is the function's derivative, with respect to the variable of interest. The integrals of a function are the components of its antiderivative. The definite integral of a function from a to b is the area of the region in the xy-plane that lies between the graph of the function and the x-axis, above the x-axis, or below the x-axis. The indefinite integral of a function is an antiderivative of the function, and can be used to find the original function when given the derivative. The definite integral of a function is a single-valued function on a given interval. It can be computed by evaluating the definite integral of a function at every x in the domain of the function, then adding the results together.

27:53

In mathematics, a technique is a method or formula for solving a problem. Techniques are often used in mathematics, physics, economics, and computer science.

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so for this problem were given N r. Intervals are 12, um, and what we have is our Delta T could also be Delta X, But this time we're talking about time is be a over end. And since we're talking about 24 hours, we'll have 24 over 12, which is to, um so since this is the Simpson's rule, we're going to have n plus one terms. So 13 terms we're going to fill in all those numbers so are integral is going to be, um, very close to Delta T over three times f of t, not plus four times f f t one plus two times f of t two plus four times f of t three. This is going to keep going on 24 to 4. And then at the very end, what we'll have is two times f of t 10 plus four times f of t 11. This is gonna be a teen at 10, um, and then plus f f t 12. So there's our 13 terms, and we know that they're going to increment. Um, based on the fact that t equals, we're gonna have t equals zero t equals two t equals for um, so it's going to increase in that way. So when we plug in all those values, what we end up getting remembering that Delta T is equal to two will end up getting approximately 17 point 101,704 0.67 So to find the average temperature, we want to take 1/24 times that value, and we see that the average temperature is 71 degrees Fahrenheit.

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