# The ceiling of a building has a height above the floor given by z = 20 + 1/4x, and one of the walls follows a path modeled by y = x^(3/2). Find the surface area of the wall if 0 ≤ x ≤ 40. (All...

The ceiling of a building has a height above the floor given by

z = 20 + 1/4x, and one of the walls follows a path modeled by y = x^(3/2). Find the surface area of the wall if 0 ≤ x ≤ 40. (All measurements are in feet. Round your answer to two decimal places.)

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The wall follows the path

`y = x^(3/2)`

and on that path has height

`z = 20 + 1/4 x`

To calculate the surface area of the wall from floor to roof, we need to integrate the length fragments of the wall multiplied by the heights at those fragments. The length of a fragment makes up the *arc length* and is given by

`dl = sqrt(1 + ((dy)/(dx))^2) dx`

at any point `(x,y)`.

We need then to integrate `z ((dl)/(dx))` with respect to `x` for `x` between 0 and 40:

` ` `int_0^40 zsqrt(1 + ((dy)/(dx))^2) dx``= int_0^40 (20+1/4x)sqrt(1+((3/2)x^(1/2))^2) dx`

```= int_0^40 (20+1/4x)3/2sqrt(4/9+x) dx`

Substituting `u = 4/9 + x` `implies` `x = u -4/9`, `(dx)/(du) = 1` we calculate

`int_(4/9)^(364/9) (179/9 + u/4)(3/2) u^(1/2) ((dx)/(du)) du = int_(4/9)^(364/9) 179/6u^(1/2)du + int_(4/9)^(364/9) 3/8u^(3/2)du`

`= (179/6)(2/3)u^(3/2)|_(4/9)^(364/9) + 6/40u^(5/2)|_(4/9)^(364/9)`

`= 179/9[ (364/9)^(3/2) - (4/9)^(3/2)] + 3/20[(364/9)^(5/2) - (4/9)^(5/2)]`

`= 5109.734 + 1560.389 = 6670.123`

**The surface area is 6670.12 square ft (to 2dp)**