Is it possible to find an example of a bounded region in the x, y plane that satisfies the following condition : when you revolve the region about the x axis you obtain a solid that has a volume equals its surface area
volume is in length^3 units, surface area is in lenght^2 units.
So no, it is not possible to equate those. Area cannot be equal to volume.
45cm^2=? 45cm^3
1=?cm No, cm cannot be unitless
Now, I assume your teacher is not considering units...just asking if the numerals can equal.
Yes, if one does not care about units, they can be "equaled", whatever that means.
It really is a stupid question, perhaps your teacher was not thinking.
yes he does not care about the unit he just want the value using integration
To find an example of a bounded region in the x, y plane that satisfies the condition of revolving the region about the x-axis to obtain a solid with a volume equal to its surface area, we can go through a step-by-step process:
Step 1: Define the region: Consider a simple shape like a circle with radius 'r' centered at the origin (0,0) in the x, y plane. This region is bounded and satisfies the condition of being a bounded region.
Step 2: Revolve the region: Next, revolve the region (circle) about the x-axis to obtain a solid. It forms a solid called a sphere.
Step 3: Calculate volume and surface area: The volume of a sphere is given by the formula V = (4/3)πr^3, and the surface area of a sphere is given by the formula SA = 4πr^2.
Step 4: Check if the volume equals the surface area: Substitute the values in the formulas. We get V = (4/3)πr^3 and SA = 4πr^2.
Now, if we set the volume equal to the surface area, we get the equation (4/3)πr^3 = 4πr^2.
Simplifying this equation, we can cancel out the common factors to get (1/3)r = 1, which gives us r = 3.
Step 5: Calculate the corresponding volume and surface area values: Substituting this value of r = 3 back into the volume and surface area formulas, we get V = (4/3)π(3^3) = 36π, and SA = 4π(3^2) = 36π.
We can observe that the volume (36π) is equal to the surface area (36π), satisfying the condition given.
Therefore, an example of a bounded region in the x, y plane that, when revolved about the x-axis, gives a solid with a volume equal to its surface area is a circle with radius 3.