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Draw the graph of $ f(x) = \sin \left (\frac{1}{2} x^2 \right) $ in the viewing rectangle $ [0, 1] $ by $ [0, 0.5] $ and let $ I = \displaystyle \int_0^1 f(x)\ dx $.(a) Use the graph to decide whether $ L_2 $, $ R_2 $, $ M_2 $, and $ T_2 $ underestimate or overestimate $ I $.(b) For any value of $ n $, list the numbers $ L_n $, $ R_n $, $ M_n $, $ T_n $, and $ I $ in increasing order.(c) Compute $ L_5 $, $ R_5 $, $ M_5 $, and $ T_5 $. From the graph, which do you think gives the best estimate of $ I $?

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a. underestimate b.$L_{n}<M_{n}<I<T_{n}<R_{n}$

Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 7

Approximate Integration

Integration Techniques

Campbell University

Baylor University

Boston College

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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Okay, problem 4 here is a graph, and then we want to decide or m and t sub 2. Let'S start 2 is just this rectangle, but there's no rectangle here and only this rectangle so is much much smaller than i and r r is the whole rectangle. Here, while it's so big and this 1, as well so or is greater than i as for m see, the function is convex, so we can conclude that m is smaller than i and t t is the error of the tribal here you can see. There is er here, so t is greater than that. So here's the answer of the part b right in part a of course l n m is underestimate, and t and r are overestimated: compute, 5 r, 5 and 5 n t 5, and you you need to know if 0.2 0.4, 0.6, 0.8 and 1 just calculate That right is not too difficult from the graph, which you think it's the best estimate of it. Of course it's between m and t right, so i can guess from the graph discuss right. Discuss, i think, is right. I just believe that m m is better than t only. This.

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