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Estimate $ \displaystyle \int_0^1 \cos (x^2)\ dx $ using (a) the Trapezoidal Rule and (b) the Midpoint Rule, each with $ n = 4 $. From a graph of the integrand, decide whether your answers are underestimates or overestimates. What can you conclude about the true value of the integral?

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Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 7

Approximate Integration

Integration Techniques

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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.

13:36

Estimate $\int_{0}^{1} \co…

07:26

14:39

(a) use the Trapezoidal Ru…

09:50

08:31

04:55

Approximate the definite i…

02:42

Approximate the following …

11:53

02:51

there are problems. Three, uh before. So you need to know. Zero two times. One over four. Just two times over too. And two times three or four. Uh, sorry. After your four and one times one bye Seems interval is one or four. You should be ready by two. Yes, there's a to here. Right? Use a calculator. You can find out there t is equal to zero point age. Nice aches. And to use a meat point rule, you need to know have one or eight and thus 3.3 over eight. And us, uh five. All right. And seven or eight. Seems the lance often throw is one over four. So they should multiply when the war for right and you can find on is equal to zero point ni oh Ni uh, from a broth, I formulated profits in Mathematica. Here scenes. The graph is concave and is decreasing. Right, So we can know that and is greater than I busies concave. And it's greater than team right? Here's a graph. He is a raft for cribs. Oi! And there's a space here and the cost. Yet I'm is greater than eyes because the function is concave, right? So what we can know? We can conclude that eyes between to appoint a United States and to appoint night all night.

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