Illustrustrate both of the following integrals. Then explain how to compute each.
\[ \int_{ 0 }^{ 5 } 5 \, \delta\left(t - 3\right) \,dt \hspace{2em} \int_{ 0 }^{ 5 } 5 \, \mathrm{u}\left(t - 3\right) \,dt \]
Answer:
\[ \int_{ 0 }^{ 5 } 5 \, \delta\left(t - 3\right) \,dt = 5 \hspace{2em} \int_{ 0 }^{ 5 } 5 \, \mathrm{u}\left(t - 3\right) \,dt = 10 \]
Illustrustrate both of the following integrals. Then explain how to compute each.
\[ \int_{ 0 }^{ 6 } 4 \, \delta\left(t - 3\right) \,dt \hspace{2em} \int_{ 0 }^{ 6 } 4 \, \mathrm{u}\left(t - 3\right) \,dt \]
Answer:
\[ \int_{ 0 }^{ 6 } 4 \, \delta\left(t - 3\right) \,dt = 4 \hspace{2em} \int_{ 0 }^{ 6 } 4 \, \mathrm{u}\left(t - 3\right) \,dt = 12 \]
Illustrustrate both of the following integrals. Then explain how to compute each.
\[ \int_{ 3 }^{ 10 } 3 \, \delta\left(t - 6\right) \,dt \hspace{2em} \int_{ 3 }^{ 10 } 3 \, \mathrm{u}\left(t - 6\right) \,dt \]
Answer:
\[ \int_{ 3 }^{ 10 } 3 \, \delta\left(t - 6\right) \,dt = 3 \hspace{2em} \int_{ 3 }^{ 10 } 3 \, \mathrm{u}\left(t - 6\right) \,dt = 12 \]
Illustrustrate both of the following integrals. Then explain how to compute each.
\[ \int_{ 4 }^{ 7 } 2 \, \delta\left(t - 5\right) \,dt \hspace{2em} \int_{ 4 }^{ 7 } 2 \, \mathrm{u}\left(t - 5\right) \,dt \]
Answer:
\[ \int_{ 4 }^{ 7 } 2 \, \delta\left(t - 5\right) \,dt = 2 \hspace{2em} \int_{ 4 }^{ 7 } 2 \, \mathrm{u}\left(t - 5\right) \,dt = 4 \]
Illustrustrate both of the following integrals. Then explain how to compute each.
\[ \int_{ 0 }^{ 5 } 4 \, \delta\left(t - 3\right) \,dt \hspace{2em} \int_{ 0 }^{ 5 } 4 \, \mathrm{u}\left(t - 3\right) \,dt \]
Answer:
\[ \int_{ 0 }^{ 5 } 4 \, \delta\left(t - 3\right) \,dt = 4 \hspace{2em} \int_{ 0 }^{ 5 } 4 \, \mathrm{u}\left(t - 3\right) \,dt = 8 \]
Illustrustrate both of the following integrals. Then explain how to compute each.
\[ \int_{ 4 }^{ 8 } 3 \, \delta\left(t - 5\right) \,dt \hspace{2em} \int_{ 4 }^{ 8 } 3 \, \mathrm{u}\left(t - 5\right) \,dt \]
Answer:
\[ \int_{ 4 }^{ 8 } 3 \, \delta\left(t - 5\right) \,dt = 3 \hspace{2em} \int_{ 4 }^{ 8 } 3 \, \mathrm{u}\left(t - 5\right) \,dt = 9 \]
Illustrustrate both of the following integrals. Then explain how to compute each.
\[ \int_{ 4 }^{ 8 } 2 \, \delta\left(t - 6\right) \,dt \hspace{2em} \int_{ 4 }^{ 8 } 2 \, \mathrm{u}\left(t - 6\right) \,dt \]
Answer:
\[ \int_{ 4 }^{ 8 } 2 \, \delta\left(t - 6\right) \,dt = 2 \hspace{2em} \int_{ 4 }^{ 8 } 2 \, \mathrm{u}\left(t - 6\right) \,dt = 4 \]
Illustrustrate both of the following integrals. Then explain how to compute each.
\[ \int_{ 3 }^{ 8 } 5 \, \delta\left(t - 5\right) \,dt \hspace{2em} \int_{ 3 }^{ 8 } 5 \, \mathrm{u}\left(t - 5\right) \,dt \]
Answer:
\[ \int_{ 3 }^{ 8 } 5 \, \delta\left(t - 5\right) \,dt = 5 \hspace{2em} \int_{ 3 }^{ 8 } 5 \, \mathrm{u}\left(t - 5\right) \,dt = 15 \]
Illustrustrate both of the following integrals. Then explain how to compute each.
\[ \int_{ 1 }^{ 6 } 2 \, \delta\left(t - 4\right) \,dt \hspace{2em} \int_{ 1 }^{ 6 } 2 \, \mathrm{u}\left(t - 4\right) \,dt \]
Answer:
\[ \int_{ 1 }^{ 6 } 2 \, \delta\left(t - 4\right) \,dt = 2 \hspace{2em} \int_{ 1 }^{ 6 } 2 \, \mathrm{u}\left(t - 4\right) \,dt = 4 \]
Illustrustrate both of the following integrals. Then explain how to compute each.
\[ \int_{ 1 }^{ 6 } 3 \, \delta\left(t - 3\right) \,dt \hspace{2em} \int_{ 1 }^{ 6 } 3 \, \mathrm{u}\left(t - 3\right) \,dt \]
Answer:
\[ \int_{ 1 }^{ 6 } 3 \, \delta\left(t - 3\right) \,dt = 3 \hspace{2em} \int_{ 1 }^{ 6 } 3 \, \mathrm{u}\left(t - 3\right) \,dt = 9 \]
Illustrustrate both of the following integrals. Then explain how to compute each.
\[ \int_{ 3 }^{ 7 } 5 \, \delta\left(t - 5\right) \,dt \hspace{2em} \int_{ 3 }^{ 7 } 5 \, \mathrm{u}\left(t - 5\right) \,dt \]
Answer:
\[ \int_{ 3 }^{ 7 } 5 \, \delta\left(t - 5\right) \,dt = 5 \hspace{2em} \int_{ 3 }^{ 7 } 5 \, \mathrm{u}\left(t - 5\right) \,dt = 10 \]
Illustrustrate both of the following integrals. Then explain how to compute each.
\[ \int_{ 2 }^{ 7 } 5 \, \delta\left(t - 5\right) \,dt \hspace{2em} \int_{ 2 }^{ 7 } 5 \, \mathrm{u}\left(t - 5\right) \,dt \]
Answer:
\[ \int_{ 2 }^{ 7 } 5 \, \delta\left(t - 5\right) \,dt = 5 \hspace{2em} \int_{ 2 }^{ 7 } 5 \, \mathrm{u}\left(t - 5\right) \,dt = 10 \]
Illustrustrate both of the following integrals. Then explain how to compute each.
\[ \int_{ 3 }^{ 9 } 3 \, \delta\left(t - 6\right) \,dt \hspace{2em} \int_{ 3 }^{ 9 } 3 \, \mathrm{u}\left(t - 6\right) \,dt \]
Answer:
\[ \int_{ 3 }^{ 9 } 3 \, \delta\left(t - 6\right) \,dt = 3 \hspace{2em} \int_{ 3 }^{ 9 } 3 \, \mathrm{u}\left(t - 6\right) \,dt = 9 \]
Illustrustrate both of the following integrals. Then explain how to compute each.
\[ \int_{ 2 }^{ 9 } 5 \, \delta\left(t - 5\right) \,dt \hspace{2em} \int_{ 2 }^{ 9 } 5 \, \mathrm{u}\left(t - 5\right) \,dt \]
Answer:
\[ \int_{ 2 }^{ 9 } 5 \, \delta\left(t - 5\right) \,dt = 5 \hspace{2em} \int_{ 2 }^{ 9 } 5 \, \mathrm{u}\left(t - 5\right) \,dt = 20 \]
Illustrustrate both of the following integrals. Then explain how to compute each.
\[ \int_{ 3 }^{ 6 } 2 \, \delta\left(t - 4\right) \,dt \hspace{2em} \int_{ 3 }^{ 6 } 2 \, \mathrm{u}\left(t - 4\right) \,dt \]
Answer:
\[ \int_{ 3 }^{ 6 } 2 \, \delta\left(t - 4\right) \,dt = 2 \hspace{2em} \int_{ 3 }^{ 6 } 2 \, \mathrm{u}\left(t - 4\right) \,dt = 4 \]
Illustrustrate both of the following integrals. Then explain how to compute each.
\[ \int_{ 4 }^{ 8 } 3 \, \delta\left(t - 5\right) \,dt \hspace{2em} \int_{ 4 }^{ 8 } 3 \, \mathrm{u}\left(t - 5\right) \,dt \]
Answer:
\[ \int_{ 4 }^{ 8 } 3 \, \delta\left(t - 5\right) \,dt = 3 \hspace{2em} \int_{ 4 }^{ 8 } 3 \, \mathrm{u}\left(t - 5\right) \,dt = 9 \]
Illustrustrate both of the following integrals. Then explain how to compute each.
\[ \int_{ 2 }^{ 5 } 2 \, \delta\left(t - 3\right) \,dt \hspace{2em} \int_{ 2 }^{ 5 } 2 \, \mathrm{u}\left(t - 3\right) \,dt \]
Answer:
\[ \int_{ 2 }^{ 5 } 2 \, \delta\left(t - 3\right) \,dt = 2 \hspace{2em} \int_{ 2 }^{ 5 } 2 \, \mathrm{u}\left(t - 3\right) \,dt = 4 \]
Illustrustrate both of the following integrals. Then explain how to compute each.
\[ \int_{ 5 }^{ 9 } 3 \, \delta\left(t - 6\right) \,dt \hspace{2em} \int_{ 5 }^{ 9 } 3 \, \mathrm{u}\left(t - 6\right) \,dt \]
Answer:
\[ \int_{ 5 }^{ 9 } 3 \, \delta\left(t - 6\right) \,dt = 3 \hspace{2em} \int_{ 5 }^{ 9 } 3 \, \mathrm{u}\left(t - 6\right) \,dt = 9 \]
Illustrustrate both of the following integrals. Then explain how to compute each.
\[ \int_{ 3 }^{ 6 } 3 \, \delta\left(t - 4\right) \,dt \hspace{2em} \int_{ 3 }^{ 6 } 3 \, \mathrm{u}\left(t - 4\right) \,dt \]
Answer:
\[ \int_{ 3 }^{ 6 } 3 \, \delta\left(t - 4\right) \,dt = 3 \hspace{2em} \int_{ 3 }^{ 6 } 3 \, \mathrm{u}\left(t - 4\right) \,dt = 6 \]
Illustrustrate both of the following integrals. Then explain how to compute each.
\[ \int_{ 3 }^{ 10 } 4 \, \delta\left(t - 6\right) \,dt \hspace{2em} \int_{ 3 }^{ 10 } 4 \, \mathrm{u}\left(t - 6\right) \,dt \]
Answer:
\[ \int_{ 3 }^{ 10 } 4 \, \delta\left(t - 6\right) \,dt = 4 \hspace{2em} \int_{ 3 }^{ 10 } 4 \, \mathrm{u}\left(t - 6\right) \,dt = 16 \]
Illustrustrate both of the following integrals. Then explain how to compute each.
\[ \int_{ 4 }^{ 9 } 4 \, \delta\left(t - 6\right) \,dt \hspace{2em} \int_{ 4 }^{ 9 } 4 \, \mathrm{u}\left(t - 6\right) \,dt \]
Answer:
\[ \int_{ 4 }^{ 9 } 4 \, \delta\left(t - 6\right) \,dt = 4 \hspace{2em} \int_{ 4 }^{ 9 } 4 \, \mathrm{u}\left(t - 6\right) \,dt = 12 \]
Illustrustrate both of the following integrals. Then explain how to compute each.
\[ \int_{ 0 }^{ 7 } 3 \, \delta\left(t - 3\right) \,dt \hspace{2em} \int_{ 0 }^{ 7 } 3 \, \mathrm{u}\left(t - 3\right) \,dt \]
Answer:
\[ \int_{ 0 }^{ 7 } 3 \, \delta\left(t - 3\right) \,dt = 3 \hspace{2em} \int_{ 0 }^{ 7 } 3 \, \mathrm{u}\left(t - 3\right) \,dt = 12 \]
Illustrustrate both of the following integrals. Then explain how to compute each.
\[ \int_{ 4 }^{ 9 } 2 \, \delta\left(t - 6\right) \,dt \hspace{2em} \int_{ 4 }^{ 9 } 2 \, \mathrm{u}\left(t - 6\right) \,dt \]
Answer:
\[ \int_{ 4 }^{ 9 } 2 \, \delta\left(t - 6\right) \,dt = 2 \hspace{2em} \int_{ 4 }^{ 9 } 2 \, \mathrm{u}\left(t - 6\right) \,dt = 6 \]
Illustrustrate both of the following integrals. Then explain how to compute each.
\[ \int_{ 1 }^{ 8 } 4 \, \delta\left(t - 4\right) \,dt \hspace{2em} \int_{ 1 }^{ 8 } 4 \, \mathrm{u}\left(t - 4\right) \,dt \]
Answer:
\[ \int_{ 1 }^{ 8 } 4 \, \delta\left(t - 4\right) \,dt = 4 \hspace{2em} \int_{ 1 }^{ 8 } 4 \, \mathrm{u}\left(t - 4\right) \,dt = 16 \]
Illustrustrate both of the following integrals. Then explain how to compute each.
\[ \int_{ 2 }^{ 7 } 5 \, \delta\left(t - 5\right) \,dt \hspace{2em} \int_{ 2 }^{ 7 } 5 \, \mathrm{u}\left(t - 5\right) \,dt \]
Answer:
\[ \int_{ 2 }^{ 7 } 5 \, \delta\left(t - 5\right) \,dt = 5 \hspace{2em} \int_{ 2 }^{ 7 } 5 \, \mathrm{u}\left(t - 5\right) \,dt = 10 \]
Illustrustrate both of the following integrals. Then explain how to compute each.
\[ \int_{ 4 }^{ 8 } 3 \, \delta\left(t - 5\right) \,dt \hspace{2em} \int_{ 4 }^{ 8 } 3 \, \mathrm{u}\left(t - 5\right) \,dt \]
Answer:
\[ \int_{ 4 }^{ 8 } 3 \, \delta\left(t - 5\right) \,dt = 3 \hspace{2em} \int_{ 4 }^{ 8 } 3 \, \mathrm{u}\left(t - 5\right) \,dt = 9 \]
Illustrustrate both of the following integrals. Then explain how to compute each.
\[ \int_{ 1 }^{ 6 } 4 \, \delta\left(t - 4\right) \,dt \hspace{2em} \int_{ 1 }^{ 6 } 4 \, \mathrm{u}\left(t - 4\right) \,dt \]
Answer:
\[ \int_{ 1 }^{ 6 } 4 \, \delta\left(t - 4\right) \,dt = 4 \hspace{2em} \int_{ 1 }^{ 6 } 4 \, \mathrm{u}\left(t - 4\right) \,dt = 8 \]
Illustrustrate both of the following integrals. Then explain how to compute each.
\[ \int_{ 4 }^{ 9 } 3 \, \delta\left(t - 6\right) \,dt \hspace{2em} \int_{ 4 }^{ 9 } 3 \, \mathrm{u}\left(t - 6\right) \,dt \]
Answer:
\[ \int_{ 4 }^{ 9 } 3 \, \delta\left(t - 6\right) \,dt = 3 \hspace{2em} \int_{ 4 }^{ 9 } 3 \, \mathrm{u}\left(t - 6\right) \,dt = 9 \]
Illustrustrate both of the following integrals. Then explain how to compute each.
\[ \int_{ 5 }^{ 9 } 4 \, \delta\left(t - 6\right) \,dt \hspace{2em} \int_{ 5 }^{ 9 } 4 \, \mathrm{u}\left(t - 6\right) \,dt \]
Answer:
\[ \int_{ 5 }^{ 9 } 4 \, \delta\left(t - 6\right) \,dt = 4 \hspace{2em} \int_{ 5 }^{ 9 } 4 \, \mathrm{u}\left(t - 6\right) \,dt = 12 \]
Illustrustrate both of the following integrals. Then explain how to compute each.
\[ \int_{ 3 }^{ 7 } 5 \, \delta\left(t - 5\right) \,dt \hspace{2em} \int_{ 3 }^{ 7 } 5 \, \mathrm{u}\left(t - 5\right) \,dt \]
Answer:
\[ \int_{ 3 }^{ 7 } 5 \, \delta\left(t - 5\right) \,dt = 5 \hspace{2em} \int_{ 3 }^{ 7 } 5 \, \mathrm{u}\left(t - 5\right) \,dt = 10 \]
Illustrustrate both of the following integrals. Then explain how to compute each.
\[ \int_{ 0 }^{ 6 } 2 \, \delta\left(t - 3\right) \,dt \hspace{2em} \int_{ 0 }^{ 6 } 2 \, \mathrm{u}\left(t - 3\right) \,dt \]
Answer:
\[ \int_{ 0 }^{ 6 } 2 \, \delta\left(t - 3\right) \,dt = 2 \hspace{2em} \int_{ 0 }^{ 6 } 2 \, \mathrm{u}\left(t - 3\right) \,dt = 6 \]
Illustrustrate both of the following integrals. Then explain how to compute each.
\[ \int_{ 1 }^{ 7 } 3 \, \delta\left(t - 3\right) \,dt \hspace{2em} \int_{ 1 }^{ 7 } 3 \, \mathrm{u}\left(t - 3\right) \,dt \]
Answer:
\[ \int_{ 1 }^{ 7 } 3 \, \delta\left(t - 3\right) \,dt = 3 \hspace{2em} \int_{ 1 }^{ 7 } 3 \, \mathrm{u}\left(t - 3\right) \,dt = 12 \]
Illustrustrate both of the following integrals. Then explain how to compute each.
\[ \int_{ 3 }^{ 8 } 4 \, \delta\left(t - 4\right) \,dt \hspace{2em} \int_{ 3 }^{ 8 } 4 \, \mathrm{u}\left(t - 4\right) \,dt \]
Answer:
\[ \int_{ 3 }^{ 8 } 4 \, \delta\left(t - 4\right) \,dt = 4 \hspace{2em} \int_{ 3 }^{ 8 } 4 \, \mathrm{u}\left(t - 4\right) \,dt = 16 \]
Illustrustrate both of the following integrals. Then explain how to compute each.
\[ \int_{ 3 }^{ 10 } 3 \, \delta\left(t - 6\right) \,dt \hspace{2em} \int_{ 3 }^{ 10 } 3 \, \mathrm{u}\left(t - 6\right) \,dt \]
Answer:
\[ \int_{ 3 }^{ 10 } 3 \, \delta\left(t - 6\right) \,dt = 3 \hspace{2em} \int_{ 3 }^{ 10 } 3 \, \mathrm{u}\left(t - 6\right) \,dt = 12 \]
Illustrustrate both of the following integrals. Then explain how to compute each.
\[ \int_{ 2 }^{ 6 } 4 \, \delta\left(t - 3\right) \,dt \hspace{2em} \int_{ 2 }^{ 6 } 4 \, \mathrm{u}\left(t - 3\right) \,dt \]
Answer:
\[ \int_{ 2 }^{ 6 } 4 \, \delta\left(t - 3\right) \,dt = 4 \hspace{2em} \int_{ 2 }^{ 6 } 4 \, \mathrm{u}\left(t - 3\right) \,dt = 12 \]
Illustrustrate both of the following integrals. Then explain how to compute each.
\[ \int_{ 4 }^{ 7 } 3 \, \delta\left(t - 5\right) \,dt \hspace{2em} \int_{ 4 }^{ 7 } 3 \, \mathrm{u}\left(t - 5\right) \,dt \]
Answer:
\[ \int_{ 4 }^{ 7 } 3 \, \delta\left(t - 5\right) \,dt = 3 \hspace{2em} \int_{ 4 }^{ 7 } 3 \, \mathrm{u}\left(t - 5\right) \,dt = 6 \]
Illustrustrate both of the following integrals. Then explain how to compute each.
\[ \int_{ 3 }^{ 10 } 2 \, \delta\left(t - 6\right) \,dt \hspace{2em} \int_{ 3 }^{ 10 } 2 \, \mathrm{u}\left(t - 6\right) \,dt \]
Answer:
\[ \int_{ 3 }^{ 10 } 2 \, \delta\left(t - 6\right) \,dt = 2 \hspace{2em} \int_{ 3 }^{ 10 } 2 \, \mathrm{u}\left(t - 6\right) \,dt = 8 \]
Illustrustrate both of the following integrals. Then explain how to compute each.
\[ \int_{ 3 }^{ 6 } 2 \, \delta\left(t - 4\right) \,dt \hspace{2em} \int_{ 3 }^{ 6 } 2 \, \mathrm{u}\left(t - 4\right) \,dt \]
Answer:
\[ \int_{ 3 }^{ 6 } 2 \, \delta\left(t - 4\right) \,dt = 2 \hspace{2em} \int_{ 3 }^{ 6 } 2 \, \mathrm{u}\left(t - 4\right) \,dt = 4 \]
Illustrustrate both of the following integrals. Then explain how to compute each.
\[ \int_{ 1 }^{ 6 } 4 \, \delta\left(t - 4\right) \,dt \hspace{2em} \int_{ 1 }^{ 6 } 4 \, \mathrm{u}\left(t - 4\right) \,dt \]
Answer:
\[ \int_{ 1 }^{ 6 } 4 \, \delta\left(t - 4\right) \,dt = 4 \hspace{2em} \int_{ 1 }^{ 6 } 4 \, \mathrm{u}\left(t - 4\right) \,dt = 8 \]
Illustrustrate both of the following integrals. Then explain how to compute each.
\[ \int_{ 3 }^{ 9 } 2 \, \delta\left(t - 5\right) \,dt \hspace{2em} \int_{ 3 }^{ 9 } 2 \, \mathrm{u}\left(t - 5\right) \,dt \]
Answer:
\[ \int_{ 3 }^{ 9 } 2 \, \delta\left(t - 5\right) \,dt = 2 \hspace{2em} \int_{ 3 }^{ 9 } 2 \, \mathrm{u}\left(t - 5\right) \,dt = 8 \]
Illustrustrate both of the following integrals. Then explain how to compute each.
\[ \int_{ 2 }^{ 6 } 3 \, \delta\left(t - 3\right) \,dt \hspace{2em} \int_{ 2 }^{ 6 } 3 \, \mathrm{u}\left(t - 3\right) \,dt \]
Answer:
\[ \int_{ 2 }^{ 6 } 3 \, \delta\left(t - 3\right) \,dt = 3 \hspace{2em} \int_{ 2 }^{ 6 } 3 \, \mathrm{u}\left(t - 3\right) \,dt = 9 \]
Illustrustrate both of the following integrals. Then explain how to compute each.
\[ \int_{ 2 }^{ 7 } 3 \, \delta\left(t - 4\right) \,dt \hspace{2em} \int_{ 2 }^{ 7 } 3 \, \mathrm{u}\left(t - 4\right) \,dt \]
Answer:
\[ \int_{ 2 }^{ 7 } 3 \, \delta\left(t - 4\right) \,dt = 3 \hspace{2em} \int_{ 2 }^{ 7 } 3 \, \mathrm{u}\left(t - 4\right) \,dt = 9 \]
Illustrustrate both of the following integrals. Then explain how to compute each.
\[ \int_{ 3 }^{ 8 } 5 \, \delta\left(t - 4\right) \,dt \hspace{2em} \int_{ 3 }^{ 8 } 5 \, \mathrm{u}\left(t - 4\right) \,dt \]
Answer:
\[ \int_{ 3 }^{ 8 } 5 \, \delta\left(t - 4\right) \,dt = 5 \hspace{2em} \int_{ 3 }^{ 8 } 5 \, \mathrm{u}\left(t - 4\right) \,dt = 20 \]
Illustrustrate both of the following integrals. Then explain how to compute each.
\[ \int_{ 1 }^{ 7 } 5 \, \delta\left(t - 4\right) \,dt \hspace{2em} \int_{ 1 }^{ 7 } 5 \, \mathrm{u}\left(t - 4\right) \,dt \]
Answer:
\[ \int_{ 1 }^{ 7 } 5 \, \delta\left(t - 4\right) \,dt = 5 \hspace{2em} \int_{ 1 }^{ 7 } 5 \, \mathrm{u}\left(t - 4\right) \,dt = 15 \]
Illustrustrate both of the following integrals. Then explain how to compute each.
\[ \int_{ 2 }^{ 8 } 2 \, \delta\left(t - 5\right) \,dt \hspace{2em} \int_{ 2 }^{ 8 } 2 \, \mathrm{u}\left(t - 5\right) \,dt \]
Answer:
\[ \int_{ 2 }^{ 8 } 2 \, \delta\left(t - 5\right) \,dt = 2 \hspace{2em} \int_{ 2 }^{ 8 } 2 \, \mathrm{u}\left(t - 5\right) \,dt = 6 \]
Illustrustrate both of the following integrals. Then explain how to compute each.
\[ \int_{ 2 }^{ 7 } 5 \, \delta\left(t - 5\right) \,dt \hspace{2em} \int_{ 2 }^{ 7 } 5 \, \mathrm{u}\left(t - 5\right) \,dt \]
Answer:
\[ \int_{ 2 }^{ 7 } 5 \, \delta\left(t - 5\right) \,dt = 5 \hspace{2em} \int_{ 2 }^{ 7 } 5 \, \mathrm{u}\left(t - 5\right) \,dt = 10 \]
Illustrustrate both of the following integrals. Then explain how to compute each.
\[ \int_{ 4 }^{ 7 } 3 \, \delta\left(t - 5\right) \,dt \hspace{2em} \int_{ 4 }^{ 7 } 3 \, \mathrm{u}\left(t - 5\right) \,dt \]
Answer:
\[ \int_{ 4 }^{ 7 } 3 \, \delta\left(t - 5\right) \,dt = 3 \hspace{2em} \int_{ 4 }^{ 7 } 3 \, \mathrm{u}\left(t - 5\right) \,dt = 6 \]
Illustrustrate both of the following integrals. Then explain how to compute each.
\[ \int_{ 0 }^{ 6 } 2 \, \delta\left(t - 3\right) \,dt \hspace{2em} \int_{ 0 }^{ 6 } 2 \, \mathrm{u}\left(t - 3\right) \,dt \]
Answer:
\[ \int_{ 0 }^{ 6 } 2 \, \delta\left(t - 3\right) \,dt = 2 \hspace{2em} \int_{ 0 }^{ 6 } 2 \, \mathrm{u}\left(t - 3\right) \,dt = 6 \]
Illustrustrate both of the following integrals. Then explain how to compute each.
\[ \int_{ 2 }^{ 6 } 3 \, \delta\left(t - 4\right) \,dt \hspace{2em} \int_{ 2 }^{ 6 } 3 \, \mathrm{u}\left(t - 4\right) \,dt \]
Answer:
\[ \int_{ 2 }^{ 6 } 3 \, \delta\left(t - 4\right) \,dt = 3 \hspace{2em} \int_{ 2 }^{ 6 } 3 \, \mathrm{u}\left(t - 4\right) \,dt = 6 \]
Illustrustrate both of the following integrals. Then explain how to compute each.
\[ \int_{ 5 }^{ 9 } 4 \, \delta\left(t - 6\right) \,dt \hspace{2em} \int_{ 5 }^{ 9 } 4 \, \mathrm{u}\left(t - 6\right) \,dt \]
Answer:
\[ \int_{ 5 }^{ 9 } 4 \, \delta\left(t - 6\right) \,dt = 4 \hspace{2em} \int_{ 5 }^{ 9 } 4 \, \mathrm{u}\left(t - 6\right) \,dt = 12 \]