12.4.5. Integration Involving Delta Function

Try these first

1. 1.

   What is the value of $\int_{-1}^{1} \delta(x) dx$?

   Hint

   Recall that the integral of the delta function over its entire range is one.

2. 2.

   If $c$ is outside the limits $a$ and $b$, what does $\int_a^b f(x) \delta(x-c) dx$ equal?

   Hint

   Think about how the sifting property works here.

3. 3.

   What is the integral of the delta function over its entire domain?

   • 0
   • 1
   • ∞

   Hint

   Remember the property of the delta function.

4. 4.

   True or False: The Dirac delta function has non-zero values everywhere.

   • True
   • False

   Hint

   Think about its definition.

5. 5.

   Evaluate the integral $\int_{-2}^{2} 3 \delta(x-1) dx$. What is the result and why?

   Hint

   Use the sifting property for this calculation.

6. 6.

   Discuss the implications of modeling an instantaneous load on a beam using the Dirac delta function. How does this simplify our calculations?

   Hint

   Think through how such conditions would change typical load evaluations.