Final answer:
The question asks for the surface area of a surface of revolution created by rotating the given curve around the x-axis. The equation of the curve is rearranged to solve for y, the derivative dy/dx is found, and the surface area formula is applied with integration from x=2 to x=7 to find the result.
Step-by-step explanation:
The student is asking about finding the area of a surface of revolution, generated by rotating a given curve about the x-axis. In this particular problem, the equation given is 25x = y² + 50, and the rotation is performed from x=2 to x=7. To calculate the surface area, we need to use the formula for the surface area of a surface of revolution, which, when rotating about the x-axis, is:
A = 2π ∫ y ∙ sqrt(1 + (dy/dx)²) dx
We first need to express y in terms of x, so rearrange the given equation as y = sqrt(25x - 50). We then find the derivative of y with respect to x, which is dy/dx = 25/(2sqrt(25x - 50)). Plugging this into our formula and integrating from x=2 to x=7 will give us the surface area.
It's essential to proceed with the integral computation to find the exact surface area.