find the volume of the solid of revolution obtained by revolving the region bounded above by the
curve y=f(x) = √16-x^2 and below by the curve y=g(x) from x=0 to x=x√2 about the x-axis
To find the volume of the solid of revolution, you can use the method of cylindrical shells. Here are the steps to calculate it:
1. Visualize the region: Sketch the curves y = f(x) and y = g(x) on a graph to understand the region being revolved.
2. Identify intersection points: Find the points where the curves intersect to determine the limits of integration. In this case, the curves intersect at x = 0 and x = √2.
3. Set up the integral: The volume of the solid of revolution can be calculated using the integral:
V = ∫[a, b] 2πx(f(x) - g(x)) dx
Here, a and b are the limits of integration, and f(x) - g(x) gives the height of the shell at each x-coordinate.
4. Substitute the equations: In this problem, the equations are y = √16 - x^2 and y = 0, so we need to express y in terms of x.
Thus, the equation becomes:
V = ∫[0, √2] 2πx(√16 - x^2 - 0) dx
5. Simplify the equation: Square the expression inside the square root and simplify further:
V = ∫[0, √2] 2πx(4 - x^2) dx
6. Evaluate the integral: Use integration techniques, such as u-substitution or expanding the expression, to integrate the equation:
V = 2π ∫[0, √2] (4x - x^3) dx
Evaluate the integral and plug in the limits of integration:
V = 2π [(2x^2) - (x^4/4)] |[0, √2]
V = 2π [2(2) - (√2)^4/4 - (0)].
7. Simplify the equation: Evaluate the expression with the limits of integration:
V = 2π [4 - 2/4]
V = 2π [4 - 1/2]
V = 7π cubic units.
Therefore, the volume of the solid of revolution is 7π cubic units.