LifeIsADangerousGame
  • LifeIsADangerousGame
Calculus help again?
Mathematics
  • Stacey Warren - Expert brainly.com
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katieb
  • katieb
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LifeIsADangerousGame
  • LifeIsADangerousGame
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LifeIsADangerousGame
  • LifeIsADangerousGame
Here's what I've done so far, but I'm not sure if it is correct. |dw:1449871443381:dw| Am I close/on the right track?
LifeIsADangerousGame
  • LifeIsADangerousGame
There should be dy on both, but I forgot them .-.

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mathmale
  • mathmale
and that of the right end of each strip \[x _{2}=f _{2}(y).\]
mathmale
  • mathmale
The easiest way to do this problem is to imagine that the area shown is divided into multiple horizontal strips. The x-coordinate of left end of each strip will then be x1=f1(y).
mathmale
  • mathmale
Look at the graph carefully. On the left of the shaded area you see a curve, a parabola. Which of the given equations describes that parabola? Solve that equation for x; the result is your \[x _{1}=f _{1}(y).\]
mathmale
  • mathmale
On the right you see a straight line. Which equation describes it? Solve that equation for x. The result is your \[x _{2}=f _{2}(y)\]
mathmale
  • mathmale
Now you can write the expression \[x _{2}-x _{1}\]
mathmale
  • mathmale
to denote the length of any one horizontal strip. Each strip has the thickess dy. what is the area of a typical strip? Hint: area=length*width. Please write out da = length * width, in terms of dy and the 2 x-coordinates.
mathmale
  • mathmale
Once you have this formula for da, integrate it. The lower limit is -1 and the upper limit is 2. If the reason for this is unclear, ask questions!
LifeIsADangerousGame
  • LifeIsADangerousGame
Okay, I'm trying to follow along. For solving the equations for x, aren't they already solved? But I know that the parabola is x = y^2 and the straight line is x = y + 2. So, (y + 2) - y^2 for the x_2 - x_1 expression? da = y + 2 * y^2 dy? I'm not sure I understand the length*width part..
LifeIsADangerousGame
  • LifeIsADangerousGame
@mathmale
IrishBoy123
  • IrishBoy123
@LifeIsADangerousGame what you did at the start looks right but sort out the limits they should be the same doing it *dy* is smart
LifeIsADangerousGame
  • LifeIsADangerousGame
Do they need to be the same limits?
LifeIsADangerousGame
  • LifeIsADangerousGame
I have to be right back...
Zarkon
  • Zarkon
I could put a box around the shaded area that has less area than 224/9
IrishBoy123
  • IrishBoy123
"Do they need to be the same limits?" yes \(\int\limits\limits_{y=-1}^{2} \; dy \qquad ( y + 2 ) - y ^2 \) https://www.wolframalpha.com/input/?i=int_%7B-1%7D%5E%7B2%7D+++%28+y+%2B+2+%29+-+y+%5E2+++dy "your" integral with limits lined up do you agree with this?
mathmale
  • mathmale
sorry for having been absent. You're right: you have 2 equations, both of which have already been solved for x. Afraid you've written them in the wrong order. The length of each horiz. strip is equal to the x value on the right, less the x-value on the left, which comes out to L = (y-2)-y^2. Dangerous Game: You were right all along. Okay, I'm trying to follow along. For solving the equations for x, aren't they already solved? But I know that the parabola is x = y^2 and the straight line is x = y + 2. So, (y + 2) - y^2 for the x_2 - x_1 expression? YES!
mathmale
  • mathmale
Do you agree that the limits of integration on y are -1 to 2? If not, what do you think the limits are? Explain. This is very important. Now write the integral that will give you the area between the two curves. It is:\[A=\int\limits_{-1}^{2}[y-2-y^2]dy\]
mathmale
  • mathmale
Find this area.
LifeIsADangerousGame
  • LifeIsADangerousGame
@mathmale the limits -1 to 2 make sense. I'd been looking for the limits for x and not y. But I'm not sure why we're looking for the limits on y. Why is it y - 2 and not y + 2 for the integral?
mathmale
  • mathmale
My mistake; apologies. It is indeed y+2, not y -2. ;( By using formulas for x in terms of y, we are eliminating the variable x, right? That's why we need limits on y, and not limits on x.
LifeIsADangerousGame
  • LifeIsADangerousGame
Oh..okay. I'm not understanding just yet, but maybe I'll get it later on. I finished it and I got 3.5 as my answer
mathmale
  • mathmale
I've done the problem and obtain 4.5 as the area. If and when you have the time and interest, contact me and I'll help you through this problem again. Thanks for your persistence.
LifeIsADangerousGame
  • LifeIsADangerousGame
Thank you for your help!

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