# Calculus 2 : Fundamental Theorem of Calculus

## Example Questions

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### Example Question #169 : Integrals

Given that , determine:

Explanation:

Since , we know that

By the fundamental theorem of calculus:

### Example Question #170 : Integrals

Find  of

Explanation:

This is a Second Fundamental Theorem of Calculus.  Since derivatives and anti-derivatives annihilate each other, we simply need to plug in the bounds into the function and multiply each by their derivative, respectively.

The second term drops out since the derivative of zero is zero.

In the first term, we again have two functions that annihilate each other:

### Example Question #21 : Fundamental Theorem Of Calculus

Explanation:

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.

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### Example Question #22 : Fundamental Theorem Of Calculus

Differentiate:

Explanation:

Differentiate:

Use the Second Fundamental Theorem of Calculus:

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If the function  is continuous on an open interval  and if  is in , then for a function  defined by , we have

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To understand how to apply the second FTOC, notice that  in our case is a function of a function. The chainrule will therefore have to be applied as well. To see how this works, let . The function can now be written in the form:

Now we can go back to writing in terms of , the derivative is,

### Example Question #23 : 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  is a constant and  is a function that represents an external force.

Equations of the form  are  sinusoidal functions, where  is the imaginary constant. Determine the external force  needed to produce solutions  of sinusoidal behavior

Explanation:

We plug in  into our formula and determine the external force .

Taking the derivative of each side with respect to , and applying the fundamental theorem of calculus:

Solving for ,

### Example Question #24 : Fundamental Theorem Of Calculus

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  is a constant and  is a function that represents an external force.

What external force is needed in order to obtain  if

Explanation:

In order to solve this, we substitute  into our equation and solve for :

Taking the derivative of each side with respect to , we get that:

Using the fundamental theorem of calculus:

Solving for :

Since ,

Let

Find .