## D1 - Discontinuous functions and distributions

#### Example 1 π

Illustrustrate both of the following integrals. Then explain how to compute each.

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

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

#### Example 2 π

Illustrustrate both of the following integrals. Then explain how to compute each.

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

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

#### Example 3 π

Illustrustrate both of the following integrals. Then explain how to compute each.

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

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

#### Example 4 π

Illustrustrate both of the following integrals. Then explain how to compute each.

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

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

#### Example 5 π

Illustrustrate both of the following integrals. Then explain how to compute each.

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

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

#### Example 6 π

Illustrustrate both of the following integrals. Then explain how to compute each.

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

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

#### Example 7 π

Illustrustrate both of the following integrals. Then explain how to compute each.

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

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

#### Example 8 π

Illustrustrate both of the following integrals. Then explain how to compute each.

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

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

#### Example 9 π

Illustrustrate both of the following integrals. Then explain how to compute each.

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

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

#### Example 10 π

Illustrustrate both of the following integrals. Then explain how to compute each.

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

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

#### Example 11 π

Illustrustrate both of the following integrals. Then explain how to compute each.

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

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

#### Example 12 π

Illustrustrate both of the following integrals. Then explain how to compute each.

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

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

#### Example 13 π

Illustrustrate both of the following integrals. Then explain how to compute each.

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

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

#### Example 14 π

Illustrustrate both of the following integrals. Then explain how to compute each.

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

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

#### Example 15 π

Illustrustrate both of the following integrals. Then explain how to compute each.

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

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

#### Example 16 π

Illustrustrate both of the following integrals. Then explain how to compute each.

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

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

#### Example 17 π

Illustrustrate both of the following integrals. Then explain how to compute each.

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

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

#### Example 18 π

Illustrustrate both of the following integrals. Then explain how to compute each.

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

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

#### Example 19 π

Illustrustrate both of the following integrals. Then explain how to compute each.

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

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

#### Example 20 π

Illustrustrate both of the following integrals. Then explain how to compute each.

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

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

#### Example 21 π

Illustrustrate both of the following integrals. Then explain how to compute each.

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

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

#### Example 22 π

Illustrustrate both of the following integrals. Then explain how to compute each.

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

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

#### Example 23 π

Illustrustrate both of the following integrals. Then explain how to compute each.

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

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

#### Example 24 π

Illustrustrate both of the following integrals. Then explain how to compute each.

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

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

#### Example 25 π

Illustrustrate both of the following integrals. Then explain how to compute each.

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

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

#### Example 26 π

Illustrustrate both of the following integrals. Then explain how to compute each.

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

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

#### Example 27 π

Illustrustrate both of the following integrals. Then explain how to compute each.

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

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

#### Example 28 π

Illustrustrate both of the following integrals. Then explain how to compute each.

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

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

#### Example 29 π

Illustrustrate both of the following integrals. Then explain how to compute each.

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

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

#### Example 30 π

Illustrustrate both of the following integrals. Then explain how to compute each.

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

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

#### Example 31 π

Illustrustrate both of the following integrals. Then explain how to compute each.

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

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

#### Example 32 π

Illustrustrate both of the following integrals. Then explain how to compute each.

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

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

#### Example 33 π

Illustrustrate both of the following integrals. Then explain how to compute each.

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

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

#### Example 34 π

Illustrustrate both of the following integrals. Then explain how to compute each.

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

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

#### Example 35 π

Illustrustrate both of the following integrals. Then explain how to compute each.

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

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

#### Example 36 π

Illustrustrate both of the following integrals. Then explain how to compute each.

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

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

#### Example 37 π

Illustrustrate both of the following integrals. Then explain how to compute each.

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

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

#### Example 38 π

Illustrustrate both of the following integrals. Then explain how to compute each.

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

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

#### Example 39 π

Illustrustrate both of the following integrals. Then explain how to compute each.

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

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

#### Example 40 π

Illustrustrate both of the following integrals. Then explain how to compute each.

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

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

#### Example 41 π

Illustrustrate both of the following integrals. Then explain how to compute each.

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

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

#### Example 42 π

Illustrustrate both of the following integrals. Then explain how to compute each.

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

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

#### Example 43 π

Illustrustrate both of the following integrals. Then explain how to compute each.

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

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

#### Example 44 π

Illustrustrate both of the following integrals. Then explain how to compute each.

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

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

#### Example 45 π

Illustrustrate both of the following integrals. Then explain how to compute each.

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

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

#### Example 46 π

Illustrustrate both of the following integrals. Then explain how to compute each.

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

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

#### Example 47 π

Illustrustrate both of the following integrals. Then explain how to compute each.

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

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

#### Example 48 π

Illustrustrate both of the following integrals. Then explain how to compute each.

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

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

#### Example 49 π

Illustrustrate both of the following integrals. Then explain how to compute each.

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

$\int_{ 2 }^{ 6 } 3 \, \delta\left(t - 4\right) \,dt = 3 \hspace{2em} \int_{ 2 }^{ 6 } 3 \, \mathrm{u}\left(t - 4\right) \,dt = 6$
$\int_{ 5 }^{ 9 } 4 \, \delta\left(t - 6\right) \,dt \hspace{2em} \int_{ 5 }^{ 9 } 4 \, \mathrm{u}\left(t - 6\right) \,dt$
$\int_{ 5 }^{ 9 } 4 \, \delta\left(t - 6\right) \,dt = 4 \hspace{2em} \int_{ 5 }^{ 9 } 4 \, \mathrm{u}\left(t - 6\right) \,dt = 12$