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Given $f(x)=\frac{2 e^{-2 x}}{1-e^{-2}}$ over the interval [0,1] . (a) Show that $f$ is a $p d f$ and determine (b) $\operatorname{Pr}(0 \leq x \leq .5),$ (c) $\operatorname{Pr}(0.5 \leq x \leq 1),$ (d) $\operatorname{Pr}(x \leq 0.5).$

(b) 0.731059(c) 0.268941(d) 0.731059

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 8

Applications of the Definite Integral

Integrals

Missouri State University

Oregon State University

Harvey Mudd College

University of Michigan - Ann Arbor

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.

04:30

Suppose that $f(x)=1.5 x^{…

01:16

$$F(x)=\left\{\begin{a…

05:18

04:55

03:05

If $\phi(x)=f(x)+f(1-x)$ a…

03:47

a. For $f(x)=x^{2} e^{x},$…

mhm Oh Mhm. So in this problem we're calculating the fx is 1.5 X square and the domain is negative ones more than X. More than one. Also them the integration is .5 X. to the power of three. That would calculate purple A zero from X. More than one. This would be F one minus F zero. Okay, let's see that. This would be .5 Problem B. This is .5 more than next. More than one. This is going to be F1 one is at .5 which is Yeah. So this would be 0.5 one minus .125. And this would be .5 times .875. And the answer would be .4 um 375. Okay. Yeah. So this will be the answers. Yeah. Mhm And for problems saying It's probability negative .5 more than X. more than .5. And the answer of the Okay, so So this is .5 Times .52 pumps three sometimes too it's going to be um Sorry, this part doesn't show up. This will be .5 to the power three which is on 125. Uh huh. For problem the we're probability acts more negative too. So this would be zero because there's no chance that it's going to be small into because xo realizing the area negative one and one. So this part is empty. So it's somebody mhm For problem me. Uh huh. We have that For bill your ex more than zero or head square inactive .5. Yeah. Once again we checked the domain. So this is domain -1 & one. So now we're going to calculate at X is either small and zero. Either greater negative .5. So there are two domains. So one is this part The other is this 1? Yeah, we can see that they overlapped for this part and the intersection of them. So The cover of it is actually greater than one. So it's always sure that Either the value square the negative .5 or it's more than zero. So it's going to be one. Mhm. The thing is that if you do the math this probability would be greater than one but probably cannot be greater than once. So it's always true that there's two equality inequality. Hold probably F. We have the probability that um so this is the Mhm. Let me check. So this would be thanks. So negative was more than X more than the green. X. It's gonna be pine 05 calculating and it will get 0.5 extra pounds three plus 30.5 is 105. So we get x square is- .9. So access the negative cubic root of .9 and the answer would be negative point 9655. So that's it are

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