Final answer:
The volume of the solid formed by rotating the region enclosed by the curves y = x^2, y = 6 - x, and x = 0 around the line y = 10 is found by first identifying the bounds of integration through the points of intersection, and then integrating the area of the washer formed at each slice between x = 0 and x = 2 using the washer method.
Step-by-step explanation:
The question asks us to find the volume of the solid formed when the region enclosed by the graphs of y = x^2, y = 6 - x, and x = 0 is rotated about the line y = 10. To do this, we first need to find the intersecting points of the curves y = x^2 and y = 6 - x to determine the limits of integration.
Setting the equations equal to each other:
x^2 = 6 - x
x^2 + x - 6 = 0
This quadratic equation factors to (x + 3)(x - 2) = 0, giving us x = -3 (which we discard because we are only considering x ≥0) and x = 2. Thus, the points of intersection are at x = 0 and x = 2.
Now, to find the volume when this region is rotated about the line y = 10, we use the washer method, which involves finding the volume of a washer (the area between two concentric circles) and integrating from x = 0 to x = 2.
The outer radius is fixed and equal to 10 - (6 - x) = x + 4, and the inner radius is 10 - x^2. The formula for the volume of the washer is π(outside radius)^2 - π(inside radius)^2. Therefore, the volume V is:
∫_0^2 π[(x + 4)^2 - (10 - x^2)^2] dx
This is an integral that can typically be evaluated to provide the volume of the solid.