Evaluating the Integral ∫∫xexy dxdy from 0≤x≤1, 0≤y≤1

  • Thread starter says
  • Start date
  • Tags
    Integral
In this case, it would be easiest to integrate with respect to x first. Then use integration by parts to solve the resulting integral. After evaluating, you should get an answer of e-2. In summary, when evaluating the integral ∫∫xexy dxdy with limits 0≤x≤1 and 0≤y≤1, using integration by parts and solving with respect to x first, the answer is e-2. The two statements (ey(y-1)+1)/y2 and (ey - (ey (y-1))/y) - 1/y are equivalent.
  • #1
says
594
12

Homework Statement


∫∫xexy dxdy where upper and lower limits are 0≤x≤1, 0≤y≤1
so I complete u substitution and get the integral
1/y2 ∫ u*eu du
Now with integration by parts I end up with
xy*exy-exy/y2
I have to evaluate this integral at 0≤x≤1, 0≤y≤1, as mentioned above.
The problem I have is that after evaluating I get an answer of 1, but I've computed this question into symbolab online and it says the answer is e-2.
I was wondering how these two statements are equivalent (below)
(ey(y-1)+1)/y2

(ey - (ey (y-1))/y) - 1/y

Homework Equations

The Attempt at a Solution

 
Physics news on Phys.org
  • #2
You do not need to use variable substitution to solve this integral. Simply set up the problem as an iterated integral. Then figure out which variable is going to be easiest to start with.
 

FAQ: Evaluating the Integral ∫∫xexy dxdy from 0≤x≤1, 0≤y≤1

1. What is the purpose of evaluating the integral?

The purpose of evaluating the integral is to find the area under the curve of a given function within a specific region. It is a way to measure the total value or quantity represented by the function.

2. How is the integral evaluated?

The integral is evaluated by using techniques such as substitution, integration by parts, and partial fractions. In this specific example, the integral is evaluated using double integration, which involves integrating with respect to both x and y variables.

3. Why are the limits of integration from 0 to 1?

The limits of integration are determined by the specified region, in this case, the square with sides of length 1. The limits of integration define the boundaries within which the function is being evaluated.

4. What is the significance of the function exy in this integral?

The function exy is the integrand, which represents the rate of change of the function with respect to both x and y. In this integral, it is being multiplied by the infinitesimal area element dxdy, which allows us to calculate the total value or quantity represented by the function within the given region.

5. Can the integral be solved without using double integration?

Yes, the integral can be solved using other methods such as using a computer program or using numerical integration techniques. However, in this specific case, double integration is the most efficient way to evaluate the integral.

Similar threads

Solve the problem involving the given double integral
Replies
1
Views
869
Volume of Static Solid Using Cross-Sectional Area (Integration)
Replies
2
Views
687
Find f(x,y) given partial derivative and initial condition
Replies
6
Views
1K
Finding the area of a double integral using dxdy instead of dydx
Replies
4
Views
2K
Double Integral via Appropriate Change of Variables
Replies
3
Views
1K
How do I evaluate this integral?
Replies
6
Views
1K
Volume of solid region double integral
Replies
27
Views
2K
Determine the mean square error of a simple distribution
Replies
4
Views
652
Question re: Limits of Integration in Cylindrical Shell Equation
Replies
24
Views
2K
Obtain a nonzero solution of ##y''-4y'+x^2(y'-4y)=0## by inspection
Replies
7
Views
1K

Hot Threads

Recent Insights

Back
Top