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With a programmable calculator (or a computer), it is possible to evaluate the expressions for the sums of areas of approximating rectangles, even for large values of $ n $, using looping. (On a TI use the Is> command or a For-EndFor loop, on a Casio use Isz, on an HP or in BASIC use a FOR-NEXT loop.) Compute the sum of the areas of approximating rectangles using equal subintervals and right endpoints for $ n $ = 10, 30, 50, and 100. Then guess the value of the exact area.

The region under $ y = x^4 $ from 0 to 1

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$\sum_{i=1}^{n} f\left(a+\left(i-\frac{1}{2}\right)\left(\frac{b-a}{n}\right)\right)$

Calculus 1 / AB

Chapter 5

Integrals

Section 1

Areas and Distances

Integration

Baylor University

University of Michigan - Ann Arbor

University of Nottingham

Idaho State University

Lectures

05:53

In mathematics, an indefinite integral is an integral whose integrand is not known in terms of elementary functions. An indefinite integral is usually encountered when integrating functions that are not elementary functions themselves.

40:35

In mathematics, integration is one of the two main operations of calculus, with its inverse operation, differentiation, being the other. Given a function of a real variable (often called "the integrand"), an antiderivative is a function whose derivative is the given function. The area under a real-valued function of a real variable is the integral of the function, provided it is defined on a closed interval around a given point. It is a basic result of calculus that an antiderivative always exists, and is equal to the original function evaluated at the upper limit of integration.

02:55

With a programmable calcul…

02:37

06:28

05:10

02:19

04:06

$7-8$ With a programmable …

03:10

02:50

Some computer algebra syst…

09:05

02:48

03:40

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