Reverse the order of integration?

Click For Summary
SUMMARY

The discussion focuses on reversing the order of integration for the double integral originally defined as dy dx, with limits 0 ≤ x ≤ ln(6) and 1 ≤ y ≤ e^x. The proposed new order is dx dy, with limits 1 ≤ x ≤ e^y and 0 ≤ y ≤ ln(6). The correct approach involves understanding the relationship between x and y, specifically that y = e^x does not imply x = e^y, and emphasizes the importance of graphing the region to visualize the limits accurately.

PREREQUISITES
  • Understanding of double integrals in calculus
  • Familiarity with the concepts of reversing the order of integration
  • Ability to graph functions and interpret regions in the Cartesian plane
  • Knowledge of exponential functions and their properties
NEXT STEPS
  • Study the process of reversing the order of integration in double integrals
  • Learn how to graph exponential functions and their intersections
  • Explore the concept of Jacobians in changing variables in multiple integrals
  • Practice solving double integrals with varying limits of integration
USEFUL FOR

Students and educators in calculus, particularly those focusing on multivariable calculus and integration techniques, as well as anyone looking to deepen their understanding of the relationships between variables in double integrals.

turbokaz
Messages
19
Reaction score
0

Homework Statement



Reverse the order of integration?
Original has it in the form of dy dx, 0<=x<=ln6, 1<=y<=e^x.
I made it dx dy, with 1<=x<=e^y, 0<=y<=ln6
Is this right?

Homework Equations





The Attempt at a Solution

 
Physics news on Phys.org
Wow, you waited 12 whole minutes before bumping? That's a good way to get banned from this forum!

Draw a graph. x ranges between 0 and ln(6) so draw two vertical lines there. For each x[/itex], y ranges from 1 to [itex]y= e^x[/itex] do draw the horizontal line y= 1 and the [itex]y= e^x[/itex]. Notice that [itex]e^0= 1[/itex] so that forms something that looks like a kind of "right triangle" with a curved hypotenuse.

Now, what are the lowest and highest value of y in that "triangle"? Saying that x< ln 6 does NOT mean y< ln 6. y is not x!
Imagine a horizontal line at some y-value. x ranges from the left end of that line to the right end. What are those values?

[itex]y= e^x[/itex] does NOT give [itex]x= e^y[/itex]! Solve [itex]y= e^x[/itex] for x.
 

Similar threads

Volume between a region (above x-axis and below parabola) and a surface
  • · Replies 4 ·
Replies
4
Views
3K
Solve the given first order differential equation
  • · Replies 8 ·
Replies
8
Views
2K
Verify Green's Theorem in the given problem
  • · Replies 10 ·
Replies
10
Views
2K
Why can one do this "trick" in the First Order Linear D.E. method?
  • · Replies 19 ·
Replies
19
Views
3K
Determine the mean square error of a simple distribution
  • · Replies 4 ·
Replies
4
Views
1K
Integration problem using inscribed rectangles
  • · Replies 14 ·
Replies
14
Views
2K
Relationship between autonomous system and related single equation
  • · Replies 3 ·
Replies
3
Views
2K
Finding the rate of change of x in the equation
  • · Replies 8 ·
Replies
8
Views
2K
Solve the problem involving the given double integral
  • · Replies 1 ·
Replies
1
Views
2K
Line integral of a scalar function about a quadrant
  • · Replies 4 ·
Replies
4
Views
2K