Find the volume of the solid that is obtained when the region under the curve
y = 4 − x^2/6 is revolved around the y -axis between y = 0 and y = 4 .
To find the volume of the solid obtained by revolving the region under the curve around the y-axis, we can use the method of cylindrical shells.
The volume of a cylindrical shell is given by the formula:
V = 2πrhΔy
In this case, we need to integrate the volume of each shell over the specified range of y.
Step 1: Express the equation of the curve in terms of y:
To find r (the radius of the cylindrical shell), we need to isolate x in terms of y in the equation y = 4 − x^2/6.
Rearranging the equation:
y = 4 - x^2/6
6y = 24 - x^2
x^2 = 24 - 6y
x = √(24 - 6y)
Step 2: Determine the limits of integration:
The region is bounded by y = 0 and y = 4. So, the limits of integration for y are 0 and 4.
Step 3: Calculate the volume of each cylindrical shell:
Substituting the values into the volume formula, we have:
V = ∫(0 to 4) 2π(√(24 - 6y))h dy
Step 4: Determine the height of each cylindrical shell:
The height of each shell, h, is equal to the difference between the upper and lower boundaries of y. In this case, it is dy, the increment of y.
Step 5: Integrate the volume expression:
Evaluate the integral ∫(0 to 4) 2π(√(24 - 6y)) dy to find the volume of the solid.
And that's how you find the volume of the solid formed by revolving the region under the curve y = 4 − x^2/6 around the y-axis between y = 0 and y = 4 using the method of cylindrical shells.