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4cosl x (8) Evaluate JJj dx dydz by switching the order of integration. 0 2y (10 points)(Grids provided for your use. Sketch not required)...

Question

4cosl x (8) Evaluate JJj dx dydz by switching the order of integration. 0 2y (10 points)(Grids provided for your use. Sketch not required)

4cosl x (8) Evaluate JJj dx dydz by switching the order of integration. 0 2y (10 points) (Grids provided for your use. Sketch not required)



Answers

Sketch the region of integration, reverse the order of integration, and evaluate the integral.
\begin{equation}
\int_{0}^{8} \int_{\sqrt[3]{x}}^{2} \frac{d y d x}{y^{4}+1}
\end{equation}

The given double Integral is Zito eight Que Build off X Andi do do I D x two right away provided about four plus one on We have to evaluate this integral by reversing the order of integration. For that, we have to sketch the reason of integration also, so we know that these are the limits for way. That means why calls to do why Medical to Cuba Graphics Onda X is equal to zero x physicals toe eight. Now I'll be graphing this equation or a graph paper to sketch the reason off integration. So this is the Y axis on. This is a big success and we know that this line represents y equals two on Discover represents Vike walls To cure graphics on this is Ecstasy equals tow it on the reason of integration. Velvet this region. Now I'm writing a double integral by reversing the order of integration. So you get Dublin trickle one by avoided about four plus one dx dy way. Now we have to take the horizontal strip to write the limits because we're taking the X first at this five years ago Gentle strip so hair x zero on here X will be quits too, You know, hair via the question of Cove is VII's equals two Cuba door fix so they can write. Why Cuba is it cold streaks? So had come y cube Onda de while image will be between this line on this line here headed by zero on Sorry y zero and hair vice to Now I'll be evaluating this. A drill on for that. Evaluating the analytical first zero Toto, this is constant. One divide with writing about four plus one because they're integrating with respect DX So here it comes X only on limiter from zero to y cube Dr D Y. Now I look like in the limits, so we'll get zero toe to one divided by y to the power four plus one. Harry, get y Q minus 00 No need to write. Not a delay. So finally begin, Right? Why Cube divide by writing about a four plus one and integrations year Toto daughter d y. Now I'll be evaluating this in trouble by using the substitution motor. So, um, checking white power for as de so we can write four White Cube do y is equal story de I've just differentiated about the site. So when Why is Zito We get d A zero? I've just pegged the value zero here. And when Vie is too, we get T s Grubauer Fordice 16. So now I'm putting all their religion in trouble, so I'll get the double integral. Sorry, not the double single Integral zero does extend on. Here it is Y Cube derided e so we can write this thing as y Cube divide ET is equal to d d by four. So we can write y cube do as duty by four. Andi, this thing we can write d bliss one. So I'm digging one by four outside on the integration off one divide by two plus one will be culture natural log deep less one and the limits are from 0 to 16. Now we ought to just put the limits one by four Natural log 17 minus zero. Here get log and a natural level one which is zero. So we're left with only this and the value off natural looks. Aaron, 10 divided by four comes out to be as 0.7083 So finally we can conclude darted the double integral que brewed off x two on Here it is zero toe eight and function is divide the X Why did about four plus one will be equals to 0.7083

And this problem we wish to evaluate the given integral by reversing the order of integration. We had great 0 to 8 Q. Bert wider to give the extra the fourth dx dy This question tells our understanding of how to evaluate multiple integral to solve. Remember that entered an integral require us to use single variable integration techniques and steps so and step one we evaluate the integral. Step two we put the results are integral with resolve so proceeding to first sketch the region are so that we can understand how to reverse the order with the area shaded in yellow, which is in terms of X from cuba Y two and Y 0 to 8. That's reversing the order of integration. Given the functions here would simply be N Y is 02 X cube and an X zero too. That's a double integral becomes integral, 02 and zero. To execute X to the fourth. Dy dx The inner integral evaluates as white from zero X cube or giving integral zero. To execute, you get to the fourth. The X by exponential integration. That gives 1/4 X four from zero to equals even 16 minus 1/4.

In the question We have expressed the integral as a prevalent integral with order Offer integration reversed The given integral is integration from zero people into integration from one who you toe the party. Why f ex lay the eggs divine? No moving towards the solution. The given, integral and very can ask double integration over the region. B f x my dear, he has been interpreted as I self second region Using horizontal cross section. We can also interpret be as a type off first region using vertical section, therefore, double integration over the region. The F x y d A can return as integration from one to here to the power bi into invigoration from Ellen X who if x y d by d. X. So, after reversing daughter off integration, we get integration from one toe to toe the party by into integration from Ln x u s xle i d by the X and this will be different alive for the given collection and you


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