What is the Correct Order for Integrating in This Scenario?

In summary, the conversation discusses solving an integral and determining the domain for two functions. The integral is \int_R x^2 + y^2 dA and the two functions are 2x-y = 0 and x^2 - y = 0, with points of intersection at (0,0) and (2,4). The conversation then discusses switching the order of integration and determining the new domain for the two integrals. The correct domain is determined to be b = x^2 and a = 2x for the first integral and b = 0 and a = 2 for the second integral. The person also realizes they made a mistake in their integration and thanks the other person for their help.
  • #1
Pearce_09
74
0
Hello,
Consider the integral:
[tex] \int_R x^2 + y^2 dA [/tex]

with the two graphs 2x-y = 0 and [tex] x^2 [/tex] - y = 0

therefore y = [tex] x^2 [/tex] and y = 2x are the two functions
and the point of intersection is at (0,0) and (2,4)

therefore
[tex] \int { \int x^2 + y^2 dx } dy [/tex]
(a - is top point of the integral and b - is the bottom)

therfor the domain for the first integral (dx) is b = y/2 and a = [tex] y^1^/^2 [/tex]

and for the second integral (dy) is b= 0 and a = 4

but when i switch the order to """"" dy dx... i get a different #.

therefore my new a,b for the integrals are
for the first integral (dy) b = [tex] x^2 [/tex] a = 2x
for the second integral (dx) b = 0 a = 2

is my change of order correct or did i do somthing wrong??
 
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  • #2
never mind.. my change of order is correct.. i just messed up on my integration
thanks
 

FAQ: What is the Correct Order for Integrating in This Scenario?

What is integration?

Integration is a mathematical process of finding the area under a curve or the accumulation of a function over a given interval. It is the inverse of differentiation and is used in various fields such as physics, economics, and engineering.

What is the change of order in integration?

The change of order in integration refers to the process of rearranging the variables in an integral expression. This is done to make the integration process easier or to solve a particular problem. It is also known as variable substitution or integration by substitution.

How do you perform a change of order in integration?

To perform a change of order in integration, you need to substitute the variable in the integrand with a new variable. This new variable should be a function of the original variable. Then, you can use the chain rule to simplify the integral and solve it using standard integration techniques.

What are the benefits of using change of order in integration?

Change of order in integration can make the integration process easier and more manageable. It can also help solve specific integration problems that are difficult to solve using standard techniques. Additionally, it can help in finding a closed-form solution for an integral that was previously unsolvable.

Can you provide an example of change of order in integration?

Sure, for example, consider the integral ∫(x^2 + 2x + 3)dx. To solve this integral, we can use the change of order technique by substituting x^2 + 2x + 3 with a new variable u. This means that u = x^2 + 2x + 3. Then, we can find du/dx = 2x + 2, which means that dx = du/(2x + 2). Substituting these values in the integral, we get ∫(x^2 + 2x + 3)dx = ∫u(1/(2x + 2))du. This new integral is easier to solve using standard techniques, and once we have the solution, we can substitute back the value of u to get the final result.

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