Use fundamental theorem of calculus to compute definite integral

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Homework Help Overview

The problem involves using the fundamental theorem of calculus to compute a definite integral, with specific attention to the limits of integration and their dependence on a variable.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • Participants discuss the need to adjust the variable of integration and the implications of having a variable upper limit. Some question the original poster's assumptions about the limits and suggest using the chain rule.

Discussion Status

Participants are actively engaging with the original poster's attempts and providing insights on the correct application of the fundamental theorem of calculus. There is a recognition of the need to apply the chain rule due to the variable upper limit, but no explicit consensus on the final approach has been reached.

Contextual Notes

There is mention of an attachment that contains the original work, which may be relevant for understanding the specific errors made in the initial attempt.

s3a
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Homework Statement


The problem and my (incorrect) work are typed and attached as TheProblemAndMyWorkTypedUp.jpg.

Homework Equations


Integral from a to b of f(t) = F(b) – F(a)

The Attempt at a Solution


As mentioned above, my work is attached as TheProblemAndMyWorkTypedUp.jpg. (The (2 – 1)^12 part “dissappeared” because d/dx (constant) = 0.)

Could someone please help me figure out why what I did is wrong?
 

Attachments

  • TheProblemAndMyWorkTypedUp.jpg
    TheProblemAndMyWorkTypedUp.jpg
    10.7 KB · Views: 534
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Replace t with x^2; then t^2 = x^4.
 
I think you need to change t so that your upper limit is x. Then I think your argument will go through.
 
Oh, yes, I jumped the gun with assuming it was just x on the upper limit of the integral!

So, this (=the attachment of this reply) is how I do it, right?
 

Attachments

  • TheProblemAndMyWorkTypedUp.jpg
    TheProblemAndMyWorkTypedUp.jpg
    12.4 KB · Views: 522
s3a said:
Oh, yes, I jumped the gun with assuming it was just x on the upper limit of the integral!

So, this (=the attachment of this reply) is how I do it, right?

Since the upper limit is a function of x, you have to apply chain rule. Straightforward application of FTOC can only be applied when the upper limit is just x and the lower limit a constant.
 
So, is this (=the attachment of this reply) how it's done?
 

Attachments

  • TheProblemAndMyWorkTypedUp.jpg
    TheProblemAndMyWorkTypedUp.jpg
    11.4 KB · Views: 494
s3a said:
So, is this (=the attachment of this reply) how it's done?
Correct :smile:
 
Yay! Thank you! :)
 

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