Finding Surface Area In Exercises 17-20, find the area of the surface.
The portion of the paraboloid in the first octant
To calculate: The area of the surface given that is given in the question as in the first octant of the given paraboloid.
The region is denoted by .
The surface area of the region R can be calculated by,
Apply the power rule of integration, that is, .
The differentiation formula is .
The equation of the circle is , where r represents the radius.
The given function is written as .
Now, on performing partial differentiation on it with respect to x, make sure to use the formula, .
Now, with respect to y.
The following can be found,
Substitute the calculated value in the formula .
For the given region, the limits need to be converted to polar points in case of both x and y.
Then, the polar curve, and . This implies to the following,
Since the region is in the first octant, the obtained region can be described as,
The region is, then, bounded by and
Thus, on applying limits to the integral using the formula ,
We obtain the following,
On substituting .
We get .
Perform integration on first
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