Changing the Limits of Integration

In summary, when using the u-substitution method, the limits of integration must also be changed to match the new variable being integrated with respect to. This is done by plugging in the original limits into the new function and adjusting accordingly. This ensures that the final answer is correct and matches the original limits of integration. Additionally, graphing the original and new functions can help visualize the change in limits and make the process easier to understand.
  • #1
bmed90
99
0
Ive seen some example of U substitution where the limits of integration are changed for example, Say we have a particle pushed along the x-axis with force=10/(1+x)^2 and we want the work required to move it 9 ft.

so Work = the integral from [0,9] of 10/(1+x)^2(dx) U substitution gives

u=1+x
du=dx

the 10 is factored out tilll the end.

Ok so, in this example at x=0, U=1 and at x=9, U=10

Now the new limits are [1,10] instead if [0,9]. But Why?
 
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  • #2
You have limits x=0 to x=9. When you substitute u = 1 + x, you no longer integrate with respect to x. You integrate with respect to u, so you must make sure to change the limits to values of u, instead of x. Luckily you have a nice formula for u. When x=0, u = 1+0 = 1, when x=9, u = 1+9 = 10. So you now want to integrate from u=1 to u=10. Does this help?
 
  • #3
no longer integrate with respect to x --> You integrate with respect to u, so you must make sure to change the limits to values of u

Can you elaborate more on this? Get basic if you have to. If you don't want to that is also ok.
 
  • #4
The limits of integration in
[tex]\int_{x= 0}^9 \frac{dx}{(1+ x)^2}[/tex]
are x values. When you make the substituion u= 1+ x, you must change everything from "x" to "u". When x= 0, u= 1+ 0= 1 and when x= 9, u= 1+ 9= 10:

[tex]\int_{u= 1}^{10} \frac{du}{u^2}[/tex]

(It's not a bad idea to actually write the "x= " or "u= " in the integral like that. Especially when you get to double and triple integrals.)

Notice that this is -1/u so evaluating betweeen 1 and 10 gives -1/10+ 1= 9/10.

If you go back to "x" the integral is -1/(x+ 1) which you evaluate between 0 and 9: -1/(9+ 1)- (-1/(0+1)= -1/10+ 1= 9/10 again.
 
  • #5
Maybe it would also help to draw some graphs that represent the change of variables.

If you plot y = 1+x on an xy-plane, it is just a line that starts at x=1 with slope 1. Take note of what the area looks like under the plot on [0,9].

Then plot v = u on a uv-plane, which is just a line that starts at u=0 with slope 1 (it's the above graph shifted down). If we find the area under this new graph on [0,9], it will clearly be less than the area on the original graph. If we instead calculate the area on [1,10], it will then be the same. The good news is that the function u = 1+x tells us exactly how to change the limits of integration.
 

What is "Changing the Limits of Integration"?

"Changing the Limits of Integration" refers to the process of altering the boundaries of integration in a mathematical equation. This allows for the evaluation of integrals over different intervals or regions.

Why is changing the limits of integration important?

Changing the limits of integration can be useful in simplifying complex integrals, allowing for easier calculations. It also allows for the integration of functions over different intervals, which may be more relevant to the problem at hand.

How do you change the limits of integration?

To change the limits of integration, you first need to determine the new limits. This can be done by substituting the new variables or limits into the original equation. Then, you can use the appropriate integration techniques to evaluate the integral with the new limits.

Are there any limitations to changing the limits of integration?

There are some limitations to changing the limits of integration. For example, the new limits must be within the original limits of integration. Additionally, the integral must still be convergent after changing the limits.

Can changing the limits of integration affect the value of the integral?

Yes, changing the limits of integration can affect the value of the integral. This is because the new limits may cover a larger or smaller area under the curve, resulting in a different value for the integral.

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