Will post later. Did you understand the working for the question on top just now?

yes i did thank you !!

i)

y = √x + 2/√x

When y = 3,

3 = √x + 2/√x
3√x = x + 2
x - 3√x + 2 = 0
(√x - 1)(√x - 2) = 0
√x = 1 or √x = 2
x = 1 or x = 4

(Alternatively, you may choose to use a substitution u = √x to do this.

3 = u + 2/u
3u = u² + 2
u² - 3u + 2 = 0
(u - 1)(u - 2) = 0
u = 1 or u = 2
√x = 1 or √x = 2
x = 1 or x = 4 )

OR

3² = (√x + 2/√x)²
9 = (√x)² + 2√x (2/√x) + (2/√x)²
9 = x + 4 + 4/x
9x = x² + 4x + 4
x² - 5x + 4 = 0
(x - 4)(x - 1) =0
x = 4 or x = 1

Area R

= (Area of Rectangle bound by lines y = 3, x-axis,x = 1 and x = 4) - area under the curve C from x = 1 to x = 4

= Length × Breadth - ∫⁴₁ (√x + 2/√x) dx

= (4 - 1) × 3 - [⅔x³/² + 4x¹/²]⁴₁

(You should know how to integrate this, so I didn't show the intermediate steps)

= 3 × 3 - ( [⅔(4)³/² + 4(4)¹/²] - [⅔(1)³/² + 4(1)¹/²] )

= 9 - (16/3 + 8 - ⅔ - 4)
= 9 - 8⅔
= ⅓

ii)

Translated 3 units in the negative y-direction means shifting the curve downwards vertically by 3 units.

so the equation becomes

y = √x + 2/√x - 3

iii)

What we did in ii) was to bring down the curve by 3 units.

Now, the same area R is now bounded by the x-axis instead (where y = 0)

The volume of revolution from rotating R about the line y = 3 for the original curve C, is the exact same as rotating about y = 0 for the new curve.

This results in :

π ∫⁴₁ y² dx (note that this is the new y)

= π ∫⁴₁ (√x + 2/√x - 3)² dx

= π ∫⁴₁ ( (√x + 2/√x)² - 2(√x + 2/√x)(3) + 3²) dx

= π ∫⁴₁ ( (√x + 2/√x)² - 6(√x + 2/√x) + 9) dx

= π ∫⁴₁ ( x + 4 + 4/x - 6(√x + 2/√x) + 9) dx

= π ∫⁴₁ (x - 6√x + 13 - 12/√x + 4/x) dx


(Shown)

π ∫⁴₁ (x - 6√x + 13 - 12/√x + 4/x) dx

= π ∫⁴₁ (x + 13 - 6(√x + 2/√x) + 4/x) dx

= π [½x² + 13x - 6(⅔x³/² + 4x¹/²) + 4 ln x]⁴₁

(⅔x³/² + 4x¹/² obtained from previous result of integration in i)

= π ( [½(4²) + 13(4) - 6(16/3 + 8) + 4 ln 4] - [½(1²) + 13(1) - 6(⅔ + 4) + 4 ln 1] )

(numerical evaluation of ⅔x³/² + 4x¹/² for x = 4 and x = 1 is also from previous result)

= π ( (8 + 52 - 80 + 4 ln 2²) - (½ + 13 - 28) )

= π (8 ln 2 - 20 + 14½)

= π (8 ln 2 - 5½)