Question about a double integral region

In summary, the conversation discusses a problem with finding the correct solution for determining the region of integration. The individual made an error in their analysis, but after graphing the region on the xy plane, they were able to identify their mistake and find the correct solutions of x=1 and x=4. They also clarify the inequality for the region.
  • #1
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Homework Statement
look at the image
Relevant Equations
double integral
Greetings All!

I have a problem finding the correct solution at first glance

My error was to determine the region of integration , for doing so I had to the intersection between y= sqrt(x) and y=2-x

to do so
x=(2-x)^2
to find at the end that x=1 or x=5

while graphically we can see that the region start from x=0 they intersect in x=1 and never meet again!

could someone help me with my confusion ?

Thank you!
1644410525129.png

 

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  • #2
Drawing the region on xy plane would help you.
 
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  • #3
anuttarasammyak said:
Drawing the region on xy plane would help you.
yes this i what I done
I just want to know why my analitical results was wrong
 
  • #4
Where exactly was your confusion? The region looks like this:
https://www.wolframalpha.com/input?i=root(x)+<+y+<+2-x
##x## is in ##[0,1]## and ##y## in ##[0,\sqrt{2}]##.

I don't see where you got ##x=5## from. Say ##t:=\sqrt{x}##. Then ##t^2+t-2=\left(t+1/2\right)^2-(1.5)^2\leq 0## and so ##0\leq t = \sqrt{x} \leq 1.5-0.5=1##.
 
  • #5
The solutions are x=1 and x=4 but the x=4 solution is not accepted because we want 2-x to be greater than zero. Remember that the inequality is $$0\leq \sqrt x\leq y\leq 2-x$$
 
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  • #6
Delta2 said:
The solutions are x=1 and x=4 but the x=4 solution is not accepted because we want 2-x to be greater than zero. Remember that the inequality is $$0\leq \sqrt x\leq y\leq 2-x$$
thanks a million! you nail it!
 
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1. What is a double integral region?

A double integral region is a two-dimensional area on a coordinate plane that is bounded by two curves. It is used in calculus to calculate the volume under a surface or the area between two surfaces.

2. How do you determine the limits of integration for a double integral region?

The limits of integration for a double integral region are determined by the boundaries of the region. These boundaries can be expressed as equations of the curves that bound the region in terms of x and y. The limits of integration will correspond to the range of x and y values that define the region.

3. What is the difference between a single integral and a double integral?

A single integral is used to calculate the area under a curve in one dimension, while a double integral is used to calculate the volume under a surface in two dimensions. A double integral involves integrating over a region in the x-y plane, while a single integral only involves integrating along a single axis.

4. Can you use a double integral to find the volume of a three-dimensional shape?

Yes, a double integral can be used to find the volume of a three-dimensional shape by integrating over a region in the x-y plane. This is known as a triple integral, as it involves integrating over three variables (x, y, and z).

5. How do you set up a double integral for a non-rectangular region?

To set up a double integral for a non-rectangular region, you can use the method of slicing. This involves breaking up the region into smaller rectangular regions and integrating over each of these regions separately. Alternatively, you can also use a change of variables to transform the non-rectangular region into a simpler shape, such as a rectangle, and then integrate over that region.

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