Question

1. \\(\int_{-2.1}^{3.4} 0.5 \\, ds\\)

Explanation:

Step1: Recall the integral of a constant

The integral of a constant \( k \) with respect to \( s \) is \( k s + C \) (where \( C \) is the constant of integration). For definite integrals, we use the Fundamental Theorem of Calculus, which states that \( \int_{a}^{b} f(s) ds = F(b) - F(a) \), where \( F(s) \) is an antiderivative of \( f(s) \).

For \( f(s) = 0.5 \), the antiderivative \( F(s) = 0.5s \).

Step2: Apply the Fundamental Theorem of Calculus

We evaluate \( F(3.4) - F(-2.1) \).

First, calculate \( F(3.4) = 0.5 \times 3.4 = 1.7 \).

Then, calculate \( F(-2.1) = 0.5 \times (-2.1) = -1.05 \).

Now, subtract: \( F(3.4) - F(-2.1) = 1.7 - (-1.05) = 1.7 + 1.05 = 2.75 \).

Answer:

\( 2.75 \)