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Compute the surface area of the part of the cylinder x^2 + y^2 = 1, z>=0, between the planes y=0 and z=y+1
And one more!: Find the area of the triangle with vertices (1,2,0), (3,0,7), and (-1,0,0) using a surface integral. Check your answer using the cross product
I have no idea how to do these.
In the first case make sure you are familiar with the surface area formula for a function z = f(x,y)
In your first problem use the analagous formula where x = f(y,z)
In particular:
x= (1-y^2)^(1/2)
dS = [(dx/dy)^2 + (dx/dz)^2 +1]^(1/2) dzdy
By Symmetry you can do half and double it
The region of integration is the trapezoidal region in the yz plane
where z varies from 0 to y + 1 and y varies from 0 to 1.
In the second problem you should know how to find the equation of a plane using 3 points. You probably did this much earlier in the semester so if you forgot look it up.
Once you have z = f(x,y) then it is fairly easy to see the region of integration in the xy plane is the triangular region with vertices
(-1,0), (1,2), and (3,0)