Final answer:
To find the volume of the object obtained by rotating region r about the x-axis, we can use the method of cylindrical shells. The volume can be calculated by integrating the circumference of each shell multiplied by its height.
Step-by-step explanation:
Volume of the Object obtained by Rotating r about the x-axis
To find the volume of the object obtained by rotating region r about the x-axis, we can use the method of cylindrical shells. The volume can be calculated by integrating the circumference of each shell multiplied by its height.
First, we need to find the limits of integration. To do this, we set the two functions f(x) = 2sqrt(x) and g(x) = x equal to each other and solve for x. The intersection points will give us the limits of integration.
The intersection points are x = 0 and x = 4. Therefore, the limits of integration are from 0 to 4.
Next, we need to find the circumference of each shell. The circumference of a shell at a given x-value is given by 2πx.
The height of each shell is given by f(x) - g(x). Therefore, the volume of each shell is given by (2πx)(f(x) - g(x)).
Finally, we integrate the volume of each shell over the interval [0, 4] to find the exact value of the volume of the object obtained by rotating region r about the x-axis.