Final answer:
To find the volume of the solid obtained by rotating the region in the first quadrant, bounded by the graphs of y = sqrt(16x) and y = x^2/16, about the y-axis, we can use washers or cylindrical shells. By integrating the difference in areas of two circles at each value of y, we can find the volume using washers. Alternatively, by integrating the circumference of each shell multiplied by its height, we can find the volume using cylindrical shells.
Step-by-step explanation:
To find the volume V using washers, we need to integrate the difference in areas of two circles at each value of y, and then rotate it about the y-axis.
The equation for the region R is y = sqrt(16x) and y = x^2/16. To find the limits of integration, we set the two equations equal to each other and solve for x. This gives us x = 4 and x = 8. Therefore, the limits of integration are from x = 4 to x = 8.
The equation for the outer radius is R = 8 and the equation for the inner radius is r = 4. The area of each washer is given by A = pi(R^2 - r^2). Now we can integrate this expression to find the volume V: V = ∫[4,8] pi(8^2 - 4^2) dx = pi ∫[4,8] (48) dx = 48pi(x)|[4,8] = 48pi(8-4) = 192pi.
Therefore, the volume V using washers is 192pi.
To find the volume V using cylindrical shells, we need to integrate the circumference of each shell multiplied by its height. The limits of integration remain the same. The equation for the radius is r = sqrt(16x) - x^2/16. The height of each shell is given by h = x. Now we can integrate to find the volume V: V = ∫[4,8] 2pi(x)(sqrt(16x) - x^2/16) dx. Evaluating this integral gives V = 2pi ∫[4,8] (16x^(3/2) - x^3/16) dx = 2pi (8x^(5/2)/5 - x^4/64) |[4,8] = (192pi/5 - 8pi/5) - (32pi/5 - pi/5) = 184pi/5.
Therefore, the volume V using cylindrical shells is 184pi/5.