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(a) Use the Midpoint Rule and the given data to estimate the value of the integral $ \displaystyle \int_1^5 f(x)\ dx $.(b) If it is known that $ -2 \le f^{\prime \prime} (x) \le 3 $ for all $ x $, estimate the error involved in the approximation in part (a).

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Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 7

Approximate Integration

Integration Techniques

Campbell University

Baylor University

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.

09:10

$\begin{array}{l}{\text { …

03:44

(a) Use the Midpoint Rule …

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07:39

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Let $f(x)=\cos \left(x^{2}…

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Use (a) the Midpoint Rule …

problem. 31. You the meat point rule and a human data. Uh, we can't find out that third axes 0.5. But if you want to use the me point rule, you need to know, uh, one point to fight, right? But you don't know so that their head, so that the fact is not 3.5 to their next. What on? Only the to this data, This data, this data on distant eyes useful and they were always want. So, uh, dancer is just 1.5 just 2.5. Just this once. Just this one. It's just 2.9 US 3.6 plus four plus 3.9. That's the sex going by plus 7.9. It's just, uh, 14.4. Here's Alyssa 0.4. And if you know that, uh, is where it's oneof this three than K is equal to three. And you can use the the homeowner here. Be minus a is fire minus one. Right. Fire by this one. Times cake hates re Daddy by 24 I square, right? Yes, it is fair. And what is that in this four? Right, Because you divided the whole goingto four pieces. So that's a 12 and two. Ah, that's Ah won over 32 right? That's home.

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