Integration - Fundamentals Thereom Of Calculus

In summary, to solve the given problem, we can use the power rule to integrate the expression (t-2)^1/3, which is the second fundamental theorem of calculus. First, we can make the substitution u=(t-2) and then use the formula for integrating u^n to find the solution.
  • #1
1calculus1
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Homework Statement


[tex]
\int_0^3\ [/tex] (t-2)^1/3

Homework Equations



Second of Fundamental Thereom of Calculus

The Attempt at a Solution



I don't know what to do first because I'm not used to questions with square roots. Once someone help me with the beginning, I can probably do it because after that it's all the same process anyways.



Help, please?
 
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  • #2
use the power rule
(u^v)'=v*u^(v-1)*u'+u^v*log(u)*v'
or since v'=0
(u^v)'=v*u^(v-1)*u' (when v'=0)
in particular
[(t-2)^(4/3)]'=(4/3)(t-2)^(1/3)
 
  • #3
"(u^v)'=v*u^(v-1)*u'+u^v*log(u)*v'"

seems like a crazy expression :smile:

OP,
Square roots work exactly the same way.
Try simple example first:

integrate (t-2)^2
 
  • #4
rootX said:
seems like a crazy expression :smile:

OP,
Square roots work exactly the same way.
Try simple example first:

integrate (t-2)^2

May be crazy but it is true. Still I think rootX is just suggesting you try the u substitution u=(t-2) and then use the power law formula for integrals.
 
  • #5
By the way- there is NO square root in this problem!

May be just me, but please do not use "square root" for all roots! It leads to things like people saying "3 squareroot of x" when they mean "cuberoot of x" and then it's time for the old two by four to come out!

In any case, a "root" is just a power- use the power rule:

Yes, make the substitution x- a= u and then
[tex]\int u^n du= \frac{1}{n+1} u^{n+1}+ C[/tex]
 

What is the Fundamental Theorem of Calculus?

The Fundamental Theorem of Calculus states that the two basic operations of calculus, differentiation and integration, are inverses of each other. This means that if the derivative of a function is known, the original function can be found by integration, and vice versa.

What is the difference between the first and second part of the Fundamental Theorem of Calculus?

The first part of the Fundamental Theorem of Calculus deals with finding the derivative of a function that is defined by an integral. The second part deals with finding the integral of a function that is defined by its derivative.

How is the Fundamental Theorem of Calculus used in real life?

The Fundamental Theorem of Calculus is used in many real-life applications, such as in physics, engineering, and economics. It allows us to solve problems involving rates of change, areas under curves, and accumulations of quantities over time.

What are some common mistakes when applying the Fundamental Theorem of Calculus?

One common mistake is forgetting to change the limits of integration when using the second part of the theorem. Another mistake is not using the correct notation for the derivative and integral. It is important to pay attention to the variables and their bounds when applying the theorem.

Can the Fundamental Theorem of Calculus be used for all types of functions?

Yes, the Fundamental Theorem of Calculus can be applied to all continuous functions, as long as they meet the conditions for integration. This includes polynomial, trigonometric, exponential, and logarithmic functions.

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