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(a) Verify that the function defined by $f(x)=\frac{15 x}{128} \sqrt{4-x}$ over the interval[0,4] is a $p d f$ Find $(\text { b) } \operatorname{Pr}(2 \leq x \leq 3),(\text { c) } \operatorname{Pr}(x \leq 2)$Given a $p d$ defined over the interval $[a, b]$, its mean, $\mu$, is defined by $\mu=\int_{a}^{b} x f(x) d x .$ In Exercises $20-21,$ find the mean.

(b) 0.353093(c) 0.381282

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 9

Two Integration Techniques

Integrals

Missouri State University

University of Michigan - Ann Arbor

Boston College

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.

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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.

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Find the average value of …

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in this motion we have to calculate the average value of the function and the function is that Mexico was too the root of two. X plus one. Okay. And the interval is 0- four. So you to calculate the rate value and we know that to calculate the rate value. The formula is you have to integrate in the limits of one divided by d minus a. Here to B affects C. D. S. Okay. So I am using here in this formula one divide by b minus the results into one divided by four minus zero and 0 to 4. Integration of two X plus one to the power three by sorry one by two dx. Ok. No one x 4. We can say that integration all to express one to divide one by two is two way three two weeks plus one for the power three vital. And since access multiplied by two so you have also divided by Okay. And finally in this integration you have to put your you want to limit the first telephone here even say that this constant can be canceled out. You can begin to invite you and you should get The constant outside. That is one bite. Well okay now I put here Initially four than it results in me into 9 to the power the vote minus. If you put your zero then went to the power provider. No 9 to the board three x 2 results in two nine window three. That is 27 -1. Okay. and we can say that 27 -1 is Okay 26 simply and if we divide it By two then 13 and six. So I got here 13 by it by six which is the average. Bellow. Okay thank you.

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