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It is shown that the integral in (7) may be accurately approximated by $$P(x) \approx \frac{1}{2}\left\{1-\frac{1}{30}\left[7 e^{-x^{2} / 2}+16 e^{-x^{2}(2-\sqrt{2})}+\left(7+\frac{\pi}{4} x^{2}\right) e^{-x^{2}}\right]\right\}^{\frac{1}{2}}$$See Bagby, Richard, J., Calculating Normal Probabilities, American Mathematical Monthly, 102 (1995) $46-49 .$ Compare this approximation with results obtained by your calculator for various values of $x$ between 0 and 3.

$$f(v(x)) v^{\prime}(x)-f(u(x)) u^{\prime}(x)$$

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 7

Substitution and Properties of Definite Integrals

Integrals

Missouri State University

Oregon State University

University of Michigan - Ann Arbor

University of Nottingham

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.

03:21

Use the Substitution Formu…

00:32

Use a CAS double-integral …

01:49

06:52

Approximating an Integral …

03:22

08:42

02:47

06:12

10:02

Probability In Exercises 7…

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