To find the area of the region in the first quadrant that is bounded by the graph of y = e^x and the line x = 2, we first need to set up the definite integral that represents this area.
1. **Identify the bounds of integration**: Since we are interested in the region in the first quadrant bounded by y = e^x and the vertical line x = 2, we can integrate from x = 0 to x = 2.
2. **Set up the integral**: The area can be calculated using the integral of the function y = e^x from 0 to 2:
Area = ∫02 ex dx
3. **Calculate the integral**: Now, we can compute this integral:
Area = [ex] from 0 to 2
= e2 – e0
= e2 – 1
4. **Final area calculation**: The numerical value of e is approximately 2.718, so we can further approximate the area:
e2 ≈ (2.718)2 ≈ 7.389, thus:
Area ≈ 7.389 – 1 ≈ 6.389.
Therefore, the area of the region in the first quadrant bounded by the graph of y = e^x and the line x = 2 is approximately 6.389 square units.