Answer is: option1
5.613Solution:
Each subinterval is of width \( \Delta x = 1 \).
Midpoint of \( [0,1] \) is \( x_1 = 0.5 \)
Midpoint of \( [1,2] \) is \( x_2 = 1.5 \)
Midpoint of \( [2,3] \) is \( x_3 = 2.5 \)
We evaluate \( f(x) = \sqrt{1 + x^2} \)
\[ f(0.5) = \sqrt{1 + 0.5^2} = \sqrt{1.25} \approx 1.118 \] \[ f(1.5) = \sqrt{1 + 1.5^2} = \sqrt{3.25} \approx 1.803 \] \[ f(2.5) = \sqrt{1 + 2.5^2} = \sqrt{7.25} \approx 2.692 \]
Multiply each function value by \( \Delta x = 1 \), then sum:
\[ \text{Approximation} = 1 \cdot (1.118 + 1.803 + 2.692) = 5.613 \]
Option A