Calculus 2 : Introduction to Integrals

Example Questions

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Example Question #23 : 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.  .

Example Question #24 : 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 ________________________________________________________________

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 #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 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 #26 : 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 , Example Question #27 : Fundamental Theorem Of Calculus

Let Find .      Explanation:

The Fundamental Theorem of Calculus tells us that therefore, From here plug in 0 into this equation.  1 2 10 11 12 13 14 15 16 18 Next → 