Question

find the area of the following surface using the given explicit description of the surface. the cone z2=4x2+y2​, for 0≤z≤12 question content area bottom part 1 set up the surface integral for the given function over the given surface s as a double integral over a region r in the​ xy-plane. ∫∫s ds=∫∫renter your response here da ​(type an exact​ answer, using π as​ needed.)

Answer 1

The required Surface area is $\frac{65\pi \sqrt{65}-\pi }{24}$.

What is surface area ?

• The Surface area of a solid object is a measure of the total area that the face of the object occupies.
• The total Surface area is calculated by adding all the areas on the face the areas of the base, top, and side shells( sides) of the object. Volume is the quantum of space that a three dimensional object takes up.
• Surface area is the quantum of space covering the outside of a three- dimensional shape. The Surface area is the total area covered by all the faces of a 3D object. For illustration, if we need to find the volume of makeup needed to paint a cell, also the face on which the makeup will be applied is its face area. It's always measured in square units.

To find the Surface area of the cone given, we will calculate the double integral of the magnitude of the normal vector to the cone.

For this we need the following partial derivations of the function given explicitly as

$z=2\sqrt{x^2} +y^2,0\leq z\leq 4,$

√f$x^2$(x,y)+f$y^2$(x,y)+1 = √5

Thus,

Surface Area= 4π√5.

The projection of the surface onto the xy-plane(xy) is the circle centered(center) at the origin and radius given by

z=8=2($x^2+y^2$)⟹$x^2+y^2$=4⟺ radius =2

Then,

Surface area =$\frac{65\pi \sqrt{65}-\pi }{24}$

Learn more about surface area here:

brainly.com/question/76387

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