D1 - Discontinuous functions and distributions


Example 1 πŸ”—

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:

\[ \int_{ 0 }^{ 5 } 5 \, \delta\left(t - 3\right) \,dt = 5 \hspace{2em} \int_{ 0 }^{ 5 } 5 \, \mathrm{u}\left(t - 3\right) \,dt = 10 \]


Example 2 πŸ”—

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:

\[ \int_{ 0 }^{ 6 } 4 \, \delta\left(t - 3\right) \,dt = 4 \hspace{2em} \int_{ 0 }^{ 6 } 4 \, \mathrm{u}\left(t - 3\right) \,dt = 12 \]


Example 3 πŸ”—

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:

\[ \int_{ 3 }^{ 10 } 3 \, \delta\left(t - 6\right) \,dt = 3 \hspace{2em} \int_{ 3 }^{ 10 } 3 \, \mathrm{u}\left(t - 6\right) \,dt = 12 \]


Example 4 πŸ”—

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:

\[ \int_{ 4 }^{ 7 } 2 \, \delta\left(t - 5\right) \,dt = 2 \hspace{2em} \int_{ 4 }^{ 7 } 2 \, \mathrm{u}\left(t - 5\right) \,dt = 4 \]


Example 5 πŸ”—

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:

\[ \int_{ 0 }^{ 5 } 4 \, \delta\left(t - 3\right) \,dt = 4 \hspace{2em} \int_{ 0 }^{ 5 } 4 \, \mathrm{u}\left(t - 3\right) \,dt = 8 \]


Example 6 πŸ”—

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:

\[ \int_{ 4 }^{ 8 } 3 \, \delta\left(t - 5\right) \,dt = 3 \hspace{2em} \int_{ 4 }^{ 8 } 3 \, \mathrm{u}\left(t - 5\right) \,dt = 9 \]


Example 7 πŸ”—

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:

\[ \int_{ 4 }^{ 8 } 2 \, \delta\left(t - 6\right) \,dt = 2 \hspace{2em} \int_{ 4 }^{ 8 } 2 \, \mathrm{u}\left(t - 6\right) \,dt = 4 \]


Example 8 πŸ”—

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:

\[ \int_{ 3 }^{ 8 } 5 \, \delta\left(t - 5\right) \,dt = 5 \hspace{2em} \int_{ 3 }^{ 8 } 5 \, \mathrm{u}\left(t - 5\right) \,dt = 15 \]


Example 9 πŸ”—

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:

\[ \int_{ 1 }^{ 6 } 2 \, \delta\left(t - 4\right) \,dt = 2 \hspace{2em} \int_{ 1 }^{ 6 } 2 \, \mathrm{u}\left(t - 4\right) \,dt = 4 \]


Example 10 πŸ”—

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:

\[ \int_{ 1 }^{ 6 } 3 \, \delta\left(t - 3\right) \,dt = 3 \hspace{2em} \int_{ 1 }^{ 6 } 3 \, \mathrm{u}\left(t - 3\right) \,dt = 9 \]


Example 11 πŸ”—

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:

\[ \int_{ 3 }^{ 7 } 5 \, \delta\left(t - 5\right) \,dt = 5 \hspace{2em} \int_{ 3 }^{ 7 } 5 \, \mathrm{u}\left(t - 5\right) \,dt = 10 \]


Example 12 πŸ”—

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:

\[ \int_{ 2 }^{ 7 } 5 \, \delta\left(t - 5\right) \,dt = 5 \hspace{2em} \int_{ 2 }^{ 7 } 5 \, \mathrm{u}\left(t - 5\right) \,dt = 10 \]


Example 13 πŸ”—

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:

\[ \int_{ 3 }^{ 9 } 3 \, \delta\left(t - 6\right) \,dt = 3 \hspace{2em} \int_{ 3 }^{ 9 } 3 \, \mathrm{u}\left(t - 6\right) \,dt = 9 \]


Example 14 πŸ”—

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:

\[ \int_{ 2 }^{ 9 } 5 \, \delta\left(t - 5\right) \,dt = 5 \hspace{2em} \int_{ 2 }^{ 9 } 5 \, \mathrm{u}\left(t - 5\right) \,dt = 20 \]


Example 15 πŸ”—

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:

\[ \int_{ 3 }^{ 6 } 2 \, \delta\left(t - 4\right) \,dt = 2 \hspace{2em} \int_{ 3 }^{ 6 } 2 \, \mathrm{u}\left(t - 4\right) \,dt = 4 \]


Example 16 πŸ”—

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:

\[ \int_{ 4 }^{ 8 } 3 \, \delta\left(t - 5\right) \,dt = 3 \hspace{2em} \int_{ 4 }^{ 8 } 3 \, \mathrm{u}\left(t - 5\right) \,dt = 9 \]


Example 17 πŸ”—

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:

\[ \int_{ 2 }^{ 5 } 2 \, \delta\left(t - 3\right) \,dt = 2 \hspace{2em} \int_{ 2 }^{ 5 } 2 \, \mathrm{u}\left(t - 3\right) \,dt = 4 \]


Example 18 πŸ”—

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:

\[ \int_{ 5 }^{ 9 } 3 \, \delta\left(t - 6\right) \,dt = 3 \hspace{2em} \int_{ 5 }^{ 9 } 3 \, \mathrm{u}\left(t - 6\right) \,dt = 9 \]


Example 19 πŸ”—

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:

\[ \int_{ 3 }^{ 6 } 3 \, \delta\left(t - 4\right) \,dt = 3 \hspace{2em} \int_{ 3 }^{ 6 } 3 \, \mathrm{u}\left(t - 4\right) \,dt = 6 \]


Example 20 πŸ”—

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:

\[ \int_{ 3 }^{ 10 } 4 \, \delta\left(t - 6\right) \,dt = 4 \hspace{2em} \int_{ 3 }^{ 10 } 4 \, \mathrm{u}\left(t - 6\right) \,dt = 16 \]


Example 21 πŸ”—

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:

\[ \int_{ 4 }^{ 9 } 4 \, \delta\left(t - 6\right) \,dt = 4 \hspace{2em} \int_{ 4 }^{ 9 } 4 \, \mathrm{u}\left(t - 6\right) \,dt = 12 \]


Example 22 πŸ”—

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:

\[ \int_{ 0 }^{ 7 } 3 \, \delta\left(t - 3\right) \,dt = 3 \hspace{2em} \int_{ 0 }^{ 7 } 3 \, \mathrm{u}\left(t - 3\right) \,dt = 12 \]


Example 23 πŸ”—

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:

\[ \int_{ 4 }^{ 9 } 2 \, \delta\left(t - 6\right) \,dt = 2 \hspace{2em} \int_{ 4 }^{ 9 } 2 \, \mathrm{u}\left(t - 6\right) \,dt = 6 \]


Example 24 πŸ”—

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:

\[ \int_{ 1 }^{ 8 } 4 \, \delta\left(t - 4\right) \,dt = 4 \hspace{2em} \int_{ 1 }^{ 8 } 4 \, \mathrm{u}\left(t - 4\right) \,dt = 16 \]


Example 25 πŸ”—

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:

\[ \int_{ 2 }^{ 7 } 5 \, \delta\left(t - 5\right) \,dt = 5 \hspace{2em} \int_{ 2 }^{ 7 } 5 \, \mathrm{u}\left(t - 5\right) \,dt = 10 \]


Example 26 πŸ”—

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:

\[ \int_{ 4 }^{ 8 } 3 \, \delta\left(t - 5\right) \,dt = 3 \hspace{2em} \int_{ 4 }^{ 8 } 3 \, \mathrm{u}\left(t - 5\right) \,dt = 9 \]


Example 27 πŸ”—

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:

\[ \int_{ 1 }^{ 6 } 4 \, \delta\left(t - 4\right) \,dt = 4 \hspace{2em} \int_{ 1 }^{ 6 } 4 \, \mathrm{u}\left(t - 4\right) \,dt = 8 \]


Example 28 πŸ”—

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:

\[ \int_{ 4 }^{ 9 } 3 \, \delta\left(t - 6\right) \,dt = 3 \hspace{2em} \int_{ 4 }^{ 9 } 3 \, \mathrm{u}\left(t - 6\right) \,dt = 9 \]


Example 29 πŸ”—

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:

\[ \int_{ 5 }^{ 9 } 4 \, \delta\left(t - 6\right) \,dt = 4 \hspace{2em} \int_{ 5 }^{ 9 } 4 \, \mathrm{u}\left(t - 6\right) \,dt = 12 \]


Example 30 πŸ”—

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:

\[ \int_{ 3 }^{ 7 } 5 \, \delta\left(t - 5\right) \,dt = 5 \hspace{2em} \int_{ 3 }^{ 7 } 5 \, \mathrm{u}\left(t - 5\right) \,dt = 10 \]


Example 31 πŸ”—

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:

\[ \int_{ 0 }^{ 6 } 2 \, \delta\left(t - 3\right) \,dt = 2 \hspace{2em} \int_{ 0 }^{ 6 } 2 \, \mathrm{u}\left(t - 3\right) \,dt = 6 \]


Example 32 πŸ”—

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:

\[ \int_{ 1 }^{ 7 } 3 \, \delta\left(t - 3\right) \,dt = 3 \hspace{2em} \int_{ 1 }^{ 7 } 3 \, \mathrm{u}\left(t - 3\right) \,dt = 12 \]


Example 33 πŸ”—

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:

\[ \int_{ 3 }^{ 8 } 4 \, \delta\left(t - 4\right) \,dt = 4 \hspace{2em} \int_{ 3 }^{ 8 } 4 \, \mathrm{u}\left(t - 4\right) \,dt = 16 \]


Example 34 πŸ”—

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:

\[ \int_{ 3 }^{ 10 } 3 \, \delta\left(t - 6\right) \,dt = 3 \hspace{2em} \int_{ 3 }^{ 10 } 3 \, \mathrm{u}\left(t - 6\right) \,dt = 12 \]


Example 35 πŸ”—

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:

\[ \int_{ 2 }^{ 6 } 4 \, \delta\left(t - 3\right) \,dt = 4 \hspace{2em} \int_{ 2 }^{ 6 } 4 \, \mathrm{u}\left(t - 3\right) \,dt = 12 \]


Example 36 πŸ”—

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:

\[ \int_{ 4 }^{ 7 } 3 \, \delta\left(t - 5\right) \,dt = 3 \hspace{2em} \int_{ 4 }^{ 7 } 3 \, \mathrm{u}\left(t - 5\right) \,dt = 6 \]


Example 37 πŸ”—

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:

\[ \int_{ 3 }^{ 10 } 2 \, \delta\left(t - 6\right) \,dt = 2 \hspace{2em} \int_{ 3 }^{ 10 } 2 \, \mathrm{u}\left(t - 6\right) \,dt = 8 \]


Example 38 πŸ”—

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:

\[ \int_{ 3 }^{ 6 } 2 \, \delta\left(t - 4\right) \,dt = 2 \hspace{2em} \int_{ 3 }^{ 6 } 2 \, \mathrm{u}\left(t - 4\right) \,dt = 4 \]


Example 39 πŸ”—

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:

\[ \int_{ 1 }^{ 6 } 4 \, \delta\left(t - 4\right) \,dt = 4 \hspace{2em} \int_{ 1 }^{ 6 } 4 \, \mathrm{u}\left(t - 4\right) \,dt = 8 \]


Example 40 πŸ”—

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:

\[ \int_{ 3 }^{ 9 } 2 \, \delta\left(t - 5\right) \,dt = 2 \hspace{2em} \int_{ 3 }^{ 9 } 2 \, \mathrm{u}\left(t - 5\right) \,dt = 8 \]


Example 41 πŸ”—

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:

\[ \int_{ 2 }^{ 6 } 3 \, \delta\left(t - 3\right) \,dt = 3 \hspace{2em} \int_{ 2 }^{ 6 } 3 \, \mathrm{u}\left(t - 3\right) \,dt = 9 \]


Example 42 πŸ”—

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:

\[ \int_{ 2 }^{ 7 } 3 \, \delta\left(t - 4\right) \,dt = 3 \hspace{2em} \int_{ 2 }^{ 7 } 3 \, \mathrm{u}\left(t - 4\right) \,dt = 9 \]


Example 43 πŸ”—

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:

\[ \int_{ 3 }^{ 8 } 5 \, \delta\left(t - 4\right) \,dt = 5 \hspace{2em} \int_{ 3 }^{ 8 } 5 \, \mathrm{u}\left(t - 4\right) \,dt = 20 \]


Example 44 πŸ”—

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:

\[ \int_{ 1 }^{ 7 } 5 \, \delta\left(t - 4\right) \,dt = 5 \hspace{2em} \int_{ 1 }^{ 7 } 5 \, \mathrm{u}\left(t - 4\right) \,dt = 15 \]


Example 45 πŸ”—

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:

\[ \int_{ 2 }^{ 8 } 2 \, \delta\left(t - 5\right) \,dt = 2 \hspace{2em} \int_{ 2 }^{ 8 } 2 \, \mathrm{u}\left(t - 5\right) \,dt = 6 \]


Example 46 πŸ”—

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:

\[ \int_{ 2 }^{ 7 } 5 \, \delta\left(t - 5\right) \,dt = 5 \hspace{2em} \int_{ 2 }^{ 7 } 5 \, \mathrm{u}\left(t - 5\right) \,dt = 10 \]


Example 47 πŸ”—

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:

\[ \int_{ 4 }^{ 7 } 3 \, \delta\left(t - 5\right) \,dt = 3 \hspace{2em} \int_{ 4 }^{ 7 } 3 \, \mathrm{u}\left(t - 5\right) \,dt = 6 \]


Example 48 πŸ”—

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:

\[ \int_{ 0 }^{ 6 } 2 \, \delta\left(t - 3\right) \,dt = 2 \hspace{2em} \int_{ 0 }^{ 6 } 2 \, \mathrm{u}\left(t - 3\right) \,dt = 6 \]


Example 49 πŸ”—

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:

\[ \int_{ 2 }^{ 6 } 3 \, \delta\left(t - 4\right) \,dt = 3 \hspace{2em} \int_{ 2 }^{ 6 } 3 \, \mathrm{u}\left(t - 4\right) \,dt = 6 \]


Example 50 πŸ”—

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:

\[ \int_{ 5 }^{ 9 } 4 \, \delta\left(t - 6\right) \,dt = 4 \hspace{2em} \int_{ 5 }^{ 9 } 4 \, \mathrm{u}\left(t - 6\right) \,dt = 12 \]