Final answer:
To find the exact area of the surface obtained by rotating the curve about the x-axis, we can use the formula for the surface area of a curve rotated about the x-axis. The formula is given by: A = 2π ∫[a,b] y(x) sqrt(1 + (y'(x))^2) dx, where y(x) is the function representing the curve and y'(x) is the derivative of y(x) with respect to x. In this case, y(x) = sqrt(1 + e^x) and the limits of integration are from 0 to 8.
Step-by-step explanation:
To find the exact area of the surface obtained by rotating the curve about the x-axis, we can use the formula for the surface area of a curve rotated about the x-axis. The formula is given by: A = 2π ∫[a,b] y(x) sqrt(1 + (y'(x))^2) dx, where y(x) is the function representing the curve and y'(x) is the derivative of y(x) with respect to x. In this case, y(x) = sqrt(1 + e^x) and the limits of integration are from 0 to 8.
To evaluate the integral, we first need to find the derivative of y(x). The derivative of y(x) = sqrt(1 + e^x) with respect to x can be found using the chain rule. Taking the derivative, we get y'(x) = (e^x)/(2sqrt(1 + e^x)).
Substituting the expression for y(x) and y'(x) into the formula, we have A = 2π ∫[0,8] sqrt(1 + e^x) sqrt(1 + (e^x/2sqrt(1 + e^x))^2) dx.
Unfortunately, evaluating this integral analytically is quite challenging. However, it can be evaluated numerically using approximation methods such as numerical integration or numerical methods like Simpson's rule or trapezoidal rule.