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Let $R$ be the square $0 \leq x \leq 4,0 \leq y \leq 4,$ and the joint pdfover $R$ is $f(x, y)=1 / 16 .$ If $E$ is the square, $0 \leq x \leq 2,0 \leq y \leq 2,$ determine $\operatorname{Pr}(E)$.

$$1 / 4$$

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 6

Double Integrals

Partial Derivatives

Oregon State University

Harvey Mudd College

University of Michigan - Ann Arbor

University of Nottingham

Lectures

12:15

In calculus, partial derivatives are derivatives of a function with respect to one or more of its arguments, where the other arguments are treated as constants. Partial derivatives contrast with total derivatives, which are derivatives of the total function with respect to all of its arguments.

04:14

07:46

$f$ is a joint probability…

In problem. 39 if if is a joint probability dynasty function on the we want to find the following probability the probability For X to be between one and 2 And why to be between zero and 1 we can do so by double integrating this function over this area. They've been integral for nine divided by 224 to blow by Square Root of X. Y. The X devoid We integrate from 1 to 2 for X And from 0 to 141. Let's integrate the inner together first. Then it's from 0 to 1. Buying divided by 224. The integration multiplied by square. It avoid because square it avoid. Is a constant relative to the X. And the integration of corrective X. We add one to the power and void by the new book decoy. The substitute from one Sorry, from 1 to 2 equals we can avoid this. I can't take this constant with this constant to get three Divided by 100 and 12. For the integral from 0 to 1 we substitute by X equals two. Then we have two cube And we take the square root of it. It gives to Lloyd by Square root of two -3. So through x equals one gives 1 multiplied by square root of why you are equals Three multiplied by two square root of 2 -1, divided by 1. 12. The black boy the integral of square. To avoid we add one to the power. Then by by the new book, The substitute from 0 to 1 when were subsumed by Y equals one gives two thirds here. Then we can remove three with 3 And 112 Divided by two, gives 56. Then It's too much brothers. Group 2 -1, divided by 56 or equals all point or city three. This is the final answer of party. For party we want to find the probability For X to be between one and 4 And why? To be between zero and school root of X. Then we will integrate have X. And Y. Over this area. Let's imagine this area and graph it. X. is bounded between one and 4 and why is bounded between 0 & four. This is the original value. But here we want to make it bounce between zero and Square Root of X. Let's draw a square root of X. When X equals zero, we're equal zero. When we when X equals one by equals one. When X equals four, Y equals two. Then it's here. Then it's something like that. And to integrate this area from one 24 And from zero square native eggs we think square and integrated first in the direction of void. For more equals zero to y equals square root of X. Then we change the order of the limits to be D. Y. The X. And Y. From zero two square root of X. Then X From 1- four. Then we integrate in this direct from wonderful equals nine divided by 224. Blow by the integral from 1- four. When value at the end of integral. It's a square root of X. It's a it's a constant relative to boy. Then the integration of square devoid is Y. To the power of three house divided by three halfs. This substitute from zero square root of X equals nine, divided by 224. Multiply it by The integration from 1 to 4. Then we have two thirds deployed by square root of X. Multiplied by this mystery by Y equals square root of X. Then it's X. is about of half Who is about of three house. When you subscribe by X equals when you subscribe by Y equals 00 the X. Then it equals three divided by 112 deployed by The integration from month to four. We can get X to the power of half plus three divided by four. It's X is about five divided by four. Yes Equals three divided by 112 to buy by the integral of X. We add 1 to the power. Then divide by the new power Subsequute from 1 to 4 Equals three divided by 112. We substitute by X equals food. Then it's a full To the power of mind. by by four minus which received by X equals one gives one. Then it equals This gives estimate to blow it by square root of two -1. And this multiplication gives 84 in the denominator, then it equals 0.257, and this is the final answer of part B.

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