Calculus 2 : Introduction to Integrals
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Example Questions
Example Question #21 : Fundamental Theorem Of Calculus
This is a Second Fundamental Theorem of Calculus problem. Since the derivative cancels out the integral, we just need to plug in the bounds into the function (top bound - bottom bottom) and multiply each by their derivative.
Example Question #24 : Fundamental Theorem Of Calculus
Differentiate:
Differentiate:
Use the Second Fundamental Theorem of Calculus:
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If the function 
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To understand how to apply the second FTOC, notice that 
Now we can go back to writing in terms of 
Example Question #25 : Fundamental Theorem Of Calculus
In exponentially decaying systems, often times the solutions to differential equations take on the form of an integral called Duhamel's Integral. This is given by:
Where 
Equations of the form 
We plug in 
Taking the derivative of each side with respect to 
Solving for 
Example Question #171 : Integrals
In harmonic systems, often times the solutions to differential equations take on the form of an integral called Duhamel's Integral. This is given by:
Where 
What external force is needed in order to obtain 
In order to solve this, we substitute 
Taking the derivative of each side with respect to 
Using the fundamental theorem of calculus:
Solving for 
Since 
Example Question #171 : Introduction To Integrals
Let
Find 
The Fundamental Theorem of Calculus tells us that
therefore,
From here plug in 0 into this equation.
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