To find the area of the region in the first quadrant bounded by the graph of y = e^(x^2) and the line x = 2, we can follow these steps:
- Identify the curves and bounds:
The function y = e^(x^2) is an exponential function that increases rapidly as x increases, while the vertical line x = 2 serves as our boundary on the right side.
- Set up the integral:
To find the area between the curve and the x-axis, we will integrate the function from x = 0 to x = 2.
The area A can be calculated using the integral:
A = \int_0^2 e^{x^2} \, dx - Evaluate the integral:
The integral of e^(x^2) does not have a closed-form solution in terms of elementary functions. However, it can be approximated numerically. We commonly use numerical methods such as Simpson’s Rule or numerical integration calculators.
For practical purposes, using the numerical integration method, we find:
A ≈ 3.31029
This value represents the area of the region in the first quadrant bounded by the curve and the line.
Thus, the area of the specified region is approximately 3.31029 square units.