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The figure shows a pendulum with length $ L $ that makes a maximum angle $ \theta_0 $ with the vertical. Using Newton's Second Law, it can be shown that the period $ T $ (the time for one complete swing) is given by$$ T = 4 \sqrt{\frac{L}{g}} \int_0^{\frac{\pi}{2}} \frac{dx}{\sqrt{1 - k^2 \sin^2 x}} $$where $ k = \sin \left (\frac{1}{2} \theta_0 \right) $ and $ g $ is the acceleration due to gravity. If $ L = 1 m $ and $ \theta_0 = 42^\circ $, use Simpson's Rule with $ n = 10 $ to find the period.

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Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 7

Approximate Integration

Integration Techniques

Missouri State University

Harvey Mudd College

University of Michigan - Ann Arbor

Idaho State 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.

05:51

The figure shows a pendulu…

13:12

12:10

The figure shows pendulum …

09:14

Period of a pendulum A sta…

00:54

09:58

Consider a pendulum of len…

04:19

Okay, so this question wants us to approximate the period of a pendulum using this expression So T is equal to four times a squared about all over g times this elliptic integral. So to evaluate this, it wants us to use Simpson's rule with an equal to 10. But first we consult for some constants to make things simpler later. So it says that l is one meter and we know that G is 9.81 times are integral from zero to pi over too of DX over square root of one minus well, k equals sign of 42 over two degrees, which equals 0.358 four. So weaken sub that in for our value of K to get are integral. So t equals four times the square root of one over 9.1 times the integral from zero pi over too, goes Well, let's just call this nasty expression f of X. So only have to do is approximate this integral now. So to do this, it wants us to take an equal to attend. So Delta X is well, we end at pi over to when we started zero and we want 10 sub intervals. So Delta X equals pi over 20. So s a 10 equals pi over 20 divided by three. So pie over 60 times f of zero plus four times af of our first value pirate 20 plus two times f of two pi over 20 plus f of 10 pie over 20. What you'll notice is pi over too. And then we also know that l equals four times squared of one over 9.81 times s a 10 earliest. That's our approximation. So if we plug in these values for our function because remember we defined f of X earlier Here is this expression. So now we can just plug this in using our calculator and see that l is approximately equal to two points. 65 We're sorry. This should be t t should be 2.65 seconds

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