Why is the derivative of the volume of a sphere equal to the surface area?

The rate of change of the volume of the sphere is equal to the surface area of the sphere. The outside of the paint is the new boundary of the sphere, and the inside of the paint is added to the volume. This explains why the derivative (rate of change) of the volume is the surface area (SA).

Is the surface area of a cylinder the derivative of its volume?

The derivative of the volume \pi r^2h of a cylinder with radius r and height h is its lateral surface area 2\pi rh.

Why is the derivative of the area of a circle its circumference?

If you increase the radius of a circle by a tiny amount, dR, then the area increases by (2πR)(dR). . That is, the derivative of the area is just the circumference. This makes the “differential nature” of the circumference a little more obvious.

How is the formula for surface area of a cylinder derived?

The formula to calculate the total surface area of a cylinder is given as, the total surface area of cylinder = 2πr(h + r), while the curved surface area of cylinder formula is, curved/lateral surface area of cylinder = 2πrh, where ‘r’ is the radius of the base and ‘h’ is the height of the cylinder.

Why is the derivative of pi r 2?

Why is the derivative of a circle’s area its perimeter (and similarly for spheres)? When differentiated with respect to r, the derivative of πr2 is 2πr, which is the circumference of a circle. Similarly, when the formula for a sphere’s volume 43πr3 is differentiated with respect to r, we get 4πr2.

What is the relationship between surface area and volume of a cylinder?

A cylinder’s volume is π r² h, and its surface area is 2π r h + 2π r². Learn how to use these formulas to solve an example problem.

What’s the relationship between surface area and volume?

The volume is how much space is inside the shape. The surface-area-to-volume ratio tells you how much surface area there is per unit of volume. This ratio can be noted as SA:V. To find this ratio, you divide the formula for surface area by the formula for volume and then you simplify.