5. \( y=e^{-x^{2}}, \quad y=0, \quad x=0, \quad x=1 \)

We want to find the area of the region bounded by the curves   y = e^(–x²), y = 0, x = 0, and x = 1. This region lies above the x-axis and under the curve y = e^(–x²) from x = 0 to x = 1. Step 1. Write the integral for the area. The area A can be found by integrating the function with respect to x from 0 to 1:   A = ∫[0 to 1] e^(–x²) dx Step 2. Notice that the antiderivative of e^(–x²) cannot be written in terms of elementary functions. However, we can express the answer in terms of the error function, erf(x). Recall that the error function is defined as   erf(x) = (2/√π) ∫[0 to x] e^(–t²) dt Thus,   ∫[0 to 1] e^(–x²) dx = (√π/2) erf(1) Step 3. Write the final answer. The exact area is   A = (√π/2) erf(1) If you need a numerical approximation, note that erf(1) ≈ 0.8427007929. Therefore,   A ≈ (√π/2) × 0.8427007929 ≈ (1.77245385/2) × 0.8427007929 ≈ 0.7468 So, the area of the region is (√π/2) erf(1), which is approximately 0.7468 square units.