## D1 - Discontinuous functions and distributions

#### Example 1 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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

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

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