Find the exact area of the surface z = 1 + 2x + 3y + 4y2, 1 ≤ x ≤ 8, 0 ≤ y ≤ 1.

The exact area of the surface is  and this can be determined by integrating the given equation.

Given:

The exact area of the surface is given by the surface integral.

   --- (1)

where and is given by:

The cross product is given by:

<1,0,2> <0,1,3 + 8y> = <-2,-3 - 8y,1>

The integral given in expression (1) is reduced to:

Now, substitute the value of [(3+8y) = tan t] and dy becomes:

Now, the integral is reduced to:

Now, simplify the above integral in order to determine the exact area of the surface.

For more information, refer to the link given below:

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where is the figure

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step-by-step explanation: