Understanding the graph for reversing an integral

In summary: XDIn summary, the student is having difficulty understanding how to graph the integral equations. They are looking for a way to graph the limits of the integrand, but are having difficulty understanding what that looks like.
  • #1
lalah
8
0

Homework Statement


I'm trying to reverse the order of integration of
[tex]\int_0^{81}\int_{y/9}^{\sqrt{y}} dx dy[/tex]
first integral is from 0 to 81
second integral is from y/9 to [tex]\sqrt{y}[/tex]

The problem is, I identified the inequalities for the regions of integration, but I'm having problems understanding how the graph is formed.

Homework Equations




The Attempt at a Solution


I understand the inequalities are y/9 < x < [tex]\sqrt{y}[/tex]
and 0 < y <81.

But I don't understand the graph.
x-axis is 0 to 9
y-axis is 0 to 81
y = x^2
y = 9x

And since according to the homework problem I'm doing, I can't progress the problem until I understand why the graph is so.

Also, can someone help me in formating the integrals in latex?
 
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  • #2
lalah said:

Homework Statement


I'm trying to reverse the order of integration of
[tex]\int_0^{81}\int_{y/9}^{\sqrt{y}} dx dy[/tex]
first integral is from 0 to 81
second integral is from y/9 to [tex]\sqrt{y}[/tex]

The problem is, I identified the inequalities for the regions of integration, but I'm having problems understanding how the graph is formed.

Homework Equations




The Attempt at a Solution


I understand the inequalities are y/9 < x < [tex]\sqrt{y}[/tex]
and 0 < y <81.

But I don't understand the graph.
x-axis is 0 to 9
y-axis is 0 to 81
y = x^2
y = 9x

And since according to the homework problem I'm doing, I can't progress the problem until I understand why the graph is so.

Also, can someone help me in formating the integrals in latex?
Draw a picture. In particular, graph x= 1/y and [itex]x= \sqrt{y}[/itex] which are the same as the hyperbola y= 1/x and the parabola y= x2. Now one problem you have here is that those two graph cross at (1,1). The area over which you are integrating is the area from the curve y= x2 on the left to the curve y= 1/x on the right up until y= 1, then from the curve y= 1/x on the left up to the curve y= x2 on the right.

Reversing the order of integration, you will need to divide this into four integrals. To cover the lower area, below y= 1, because of the change at (1,1), you will need to integrate from 0 up to y= x2 with x from 0 to 1, then from 0 up to y= 1/x, with x from 0 to 1/81. To cover the upper area, above y= 1, you need to integrate from y= 1/x up to y= 81, with x from 1/81 to 1, then from y= x2 up to y= 81, with x from 1 to up to 9:
[tex]\int_{x=0}^1\int_{y= 0}^{x^2} dy dx+ \int_{x=1}^\infty \int{y= 0}^{1/x} dy dx+ \int_{x= 1/81}^1 \int_{y=1/x}^81 dy dx+ \int_{x=1}^9\int_{y= x^2}^{81} dy dx[/tex]
 
  • #3
HallsofIvy said:
Draw a picture. In particular, graph x= 1/y and [itex]x= \sqrt{y}[/itex] which are the same as the hyperbola y= 1/x and the parabola y= x2.
I'm suck on that step. How do you determine which formulas to graph? I'm looking at the equations and inequalities, but I can't arrive to that step.

I can do reversing of integration, but this is the pothole in the bridge.
 
  • #4
The graphs you should sketch are those determined by the limits of the integrands. The limits for x are x=y/9 and x=sqrt{y}. I think HallsOfIvy meant x=y/9 instead of 1/y.
 
  • #5
Defennder said:
The graphs you should sketch are those determined by the limits of the integrands. The limits for x are x=y/9 and x=sqrt{y}. I think HallsOfIvy meant x=y/9 instead of 1/y.
Oh yes, I get it now. Thanks! :)
 
  • #6
Right! my eyes are going!
 

1. What is an integral?

An integral is a mathematical concept that represents the area under a curve on a graph. It is used to find the total value of a function over a specific interval.

2. What does it mean to reverse an integral?

Reversing an integral means finding the original function from its derivative. This is also known as anti-differentiation or integration.

3. How do you reverse an integral?

To reverse an integral, you need to use the reverse rules of differentiation. This includes using the power rule, product rule, quotient rule, and chain rule in reverse to find the original function.

4. Why is understanding the graph important for reversing an integral?

The graph of a function provides visual representation of its behavior and can help in understanding the relationship between the function and its derivative. It also helps in identifying the key points for integration, such as the area under the curve and the points of intersection.

5. What are some common mistakes when reversing an integral?

Some common mistakes when reversing an integral include forgetting to add the constant of integration, using the wrong integration rules, and not considering the limits of integration when solving definite integrals.

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