Evaluate the integral by interpreting it in terms of areas.
\[
\int_{-1}^{0}\left(2+\sqrt{1-x^{2}}\right) d x
\]
Final Answer: The area under the curve of the function \(f(x) = 2 + \sqrt{1 - x^2}\) from \(x = -1\) to \(x = 0\) is \(\boxed{\frac{\pi}{4} + 2}\).
Step 1 :The integral can be interpreted as the area under the curve of the function \(f(x) = 2 + \sqrt{1 - x^2}\) from \(x = -1\) to \(x = 0\).
Step 2 :To find this area, we can use the definite integral, which is the fundamental concept of Calculus. The definite integral of a function can be interpreted as the signed area of the region bounded by the function and the x-axis, from \(x = a\) to \(x = b\).
Step 3 :The area under the curve of the function \(f(x) = 2 + \sqrt{1 - x^2}\) from \(x = -1\) to \(x = 0\) is calculated as \(\frac{\pi}{4} + 2\).
Step 4 :Final Answer: The area under the curve of the function \(f(x) = 2 + \sqrt{1 - x^2}\) from \(x = -1\) to \(x = 0\) is \(\boxed{\frac{\pi}{4} + 2}\).