Answer:
To find the volume of the solid obtained by rotating the region bounded by the curve y = x^2, we need to specify the axis of rotation. Let's assume we are rotating the region about the x-axis.
To find the volume, we can use the method of cylindrical shells. Here are the steps:
1. Determine the limits of integration: We need to find the x-values where the curve y = x^2 intersects the x-axis. In this case, it intersects at x = 0 and x = 1.
2. Set up the integral: The volume of a cylindrical shell is given by the formula V = 2π * ∫[a,b] (x * f(x) * dx), where a and b are the limits of integration, and f(x) is the function that represents the curve.
So, the integral for this problem would be: V = 2π * ∫[0,1] (x * x^2 * dx).
3. Evaluate the integral: We integrate the expression with respect to x. The integral of x^3 with respect to x is (1/4)x^4.
V = 2π * [(1/4)x^4] evaluated from 0 to 1.
Plugging in the values, we get V = 2π * [(1/4)(1^4) - (1/4)(0^4)].
Simplifying further, V = π/2.
4. Finalize the answer: The volume of the solid obtained by rotating the region bounded by the curve y = x^2 about the x-axis is π/2 cubic units.
Remember, this method is specific to rotating about the x-axis. If the axis of rotation is different, the steps may vary.
Explanation: