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Find $iint_{R} f(x, y) d A$ for the given functions and regions. $f(x, y)=7$ and $R=[3,5] imes[-1,9]$...

Question

Find $iint_{R} f(x, y) d A$ for the given functions and regions. $f(x, y)=7$ and $R=[3,5] imes[-1,9]$

Find $iint_{R} f(x, y) d A$ for the given functions and regions. $f(x, y)=7$ and $R=[3,5] imes[-1,9]$



Answers

Evaluate the integral by making an appropriate
change of variables.
$$\begin{array}{l}{\iint_{R} \cos \left(\frac{y-x}{y+x}\right) d A, \text { where } R \text { is the trapezoidal region }} \\ {\text { with vertices }(1,0),(2,0),(0,2), \text { and }(0,1)}\end{array}$$

We want to evaluate this integral here, so let's just go ahead and get started. So first we integrate with respect to Z, so that's just going to give us. It would be easy evaluated from zero to the square root of nine minus expert and then D Y X, and so evaluating that would just give us square root of nine minus expert. So 09 minus X squared and down here we also have the sort of nine minus expert. Then we have D Y D X. Now our function here does not depend on X it'll so when we integrate, that would really just be times why they have 0 to 3. Why times nine minus expired, evaluated from zero to the square root of nine months. Expert DX. So when we evaluate that when we first plug in this world of nine minus X squared, we would have where those two square roots would counts out and we just have nine minus sechs where when we plug in zero would just get zero. Now we can go ahead and integrate this using power rules who end up with nine x minus 1/3 X cube violet from 0 to 3. So when we plug in three wind up with 27 minus and three cute is 27 divided by three is nine. And when we plug in, zero would just get zero. So this is 18 so we should get 18 for evaluating this.

In Problem 39. We want to evaluate the double integral over the region D for this function the double integral e to the bar of X squared the ex doi that's define the X or do y we have here eat the ball off X squared. We can't evaluate it. Then we started, by the way, the X to make it as a constant for the first integral. Then we define it as a vertical segment. Take a vertical segment. This vertical segments starts at this line and ends adidas line this line the line below it has the equation. Why equals M, which is the slope of the line? The slope is D y divided by the X. The boy is one and the excess four minus or plus the Y intercept, which is you. The equation for the other line. The above line of the region equals why y equals the slope of this line. Dear boy, it's your three divided by the X, which is four multiplied by X plus y intercept, which is you, then The limits for why it starts at X, divided by four and ends at three. X, divided by four while in the horizontal direction. X starts we integrate from zero 24 equals for the inner integral. We have a constant well tabloid. The oy we have It is about X squared. That integration of the Roy is just a boy we substitute from X, divided by four three x divided by four equals e to the power of excess square. The integral The year 24 mhm to the power of X squared We my substitute by y equals three x divided by four minus. It was a lot of X squared multiplied boy X divided boy four The X equals We can take wonderful by four out of the integral and take off from 0 to 4. We have three x multiplied Boy, it's a lot of X squared. We have eaten a lot of Texas quit and it's multiplied by X or three X. If we multiplied it by two, then we have three. But the blow it by two divided by two to make this multiplication is the differential off the bar function off this polynomial to easily integrate it. The bar of extra square minus. We do the same, multiply by two and divide by two two x multiplied by each. A lot of extra square which is the integral the differential off this bower function, the X equals we can take off the out of the integral off the integral from 0 to 4 three minus one, which is one all we can evaluate. Three multiplied boy. The integral will be the theme. The same function minus would be the same function. Three minus two. We have integrated already. Then we don't have this sign. We just substitute from 0 to 4. It's substitute. We have two divided by 8 1/4 but to buoyed by E to the power of X squared, which is 16 minus e to the world of zero equals one divided by four but employed by each about 16 minus one, which is the final answer off our problems.

In problem 40. We want to evaluate the double integral off dysfunction one minus two X the x d boy over this. Do you mean this domain can be expressed as horizontal December region? Because it starts at this line and it's at this line The equation off the first line here, why equals the slope? The slope is Delta X, divided by Delta y divided by the X. We have three divided by two x plus the intercept, which is the you and this line has y equals the slope, which is divided, divided by the ex deRoy is three divided by five x plus intercept, which is you. This makes X equals two thirds avoid and this makes X equals five thirds Avoid. Then we started by D X. You are where the extra started as at this line, two thirds of boy. And in this ad, five thirds avoid while we integrate This segment from zero 23 equals the integral our integral outer integral on the inner intake. The evaluation off one the X is X minus. Integration of two x is X squared. We substitute for X from two thirds to avoid 25 thirty's avoid. Let's evaluate. Let's substitute by the upper bound we have here devoid for the other. Bound we have 5 34 minus 25 by the by nine Y squared minus minus. We subdued by the lower bound. We have two thirds boy minus for divided by nine by square. Do you want that symbol for it? We have five. We have void. This comes to be void and we have 24. 25 miles four divided by nine Gives seven. Divided by three minus seven. Body boy three. Why squared Do you want 25 minus forced into underworld, Benign. It's driven by the boy. Three equals the integral we integrate. Why Toby y squared while by two minus We have seven divided by nine. Why cube? We substitute from 0 to 3. We substitute first. Bye bye Equals three. I have line divided by two minus seven, divided by nine, multiplied by nine multiplied by three minus zero equals 4.5 minus. We have 21 minus seven multiplied by three, which is 21. It was minus 16.5, which is the final answer off our problem. The minus sign here might indicate volume below the X Y plane. There is no problem with this negative sign. And this is the final answer of all broke.

In Brooklyn. 41. We want to evaluate the double integral for the function If we're if equals X divided by I Square the ex devoid. We want to start by the X to evaluate the integral of x dx And to consider why squared is a constant with respect to X and we multiply Boy deRoy Then we would expect we would express this region as horizontal December region by taking a Rozental segment. It starts at this line where it has equation y equals the slope. The slope is deltoid, divided by Delta X. The toy is too divided by Delta X, which is two X plus the intercept the Y intercept, which is one this makes X equals Why minus one and here X starts at this line. This line equation is we have voy equals the slope, which is minus two divided by two x plus the y intercept. If we send this fine 12, the sect ads then X equals seven minus one, Then the limits off the other. Integration is seven is why my swan 27 Minus y. Let's just start the boy. The inner integral equals. We have not set the limits for why boy starts at why equals two and ends at why equals four, 2 to 4. Yeah, The integral from 2 to 4. Doi The integration of X d. X is X squared, divided by two and we have a constant in the denominator. Boy square substitute X from Y minus one, 27 minus y. Let's substitute. It equals seven minus y we have here integral seven minus y squared. Divided by why squared We can take half out of the integral minus. Why am I in this one? All squared, Divided by why is quit This is why not to do you We have the difference between two square numbers. Then we can dick. This can fact arise integral From 2 to 4 we have a voice squared here and we have seven minus y minus y last one And the other factor is seven minus y plus why minus one do you equals half but the boy boy from 2 to 4 integral from to the food we have seven minus minus y plus one. It's minus toy. Multiply it, boy seven minus one, which is six divided by y squared away can take six out of the integral equals three. The girl from 2 to 4 We have it divided by y squared minus to divided by boy New boy Then we can evaluate now the integral the integrity off Why do the bottle minus two? We had one to the bar and bar by the new bar We have it divided by y divided by minus one and she's minus it minus to the the gration. Wonderful boy is Lynn boy. We substitute from 2 to 4. Let's start by why equals for equals three multiplied by minus eight, divided by four By this to 94 minus we substitute by equals two We have minus eight, divided by two minus two Then, too, that symbol FOI equals the three. But the boy boy minus it divided by four minus two on we have plus four which is three a tabloid boy three minus two name for plus to lend to. We can rewrite rainfall as then two squared equals three multiplied boy by three This is not three. This is to we have minus to plus four which is to then we have to and we have the year minus to lend to square lost two men to. Then it finally equals three, multiplied by two minus four plus two, which is minus two, Len. To which is the final answer off our problem. Or we can simplify it to be six multiplied by one minus then, too. And this is the final answer off our problem.


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