The cylinder shell method here would be easier here instead of the disk method.


Disk method :

①At the intersection point of y = x/(x + 1) and y = -x,

-x = x/(x + 1)
-x(x + 1) = x
-x² - x = x
x² + 2x = 0
x(x + 2) = 0
x = 0 or x = -2(this one corresponds to the point of intersection)

Then, y = -(-2) = 2
So this point has coordinates (-2,2)


At the intersection of x = -4 and y = x/(x+1),
y = -4/(-4 + 1) = 4/3

So this point has coordinates (-4,4/3)


② Next, we have to integrate with respect to y. So express x in terms of y.

However, since the region is centred about x = -4, we should translate the curve 4 units to the right first, in order to integrate by revolution about the y-axis (where x = 0)

y = x/(x + 1) → y = (x - 4)/(x - 4 + 1)

y = (x - 4)/(x - 3)
y = 1 - 1/(x - 3)
1/(x - 3) = 1 - y
1/(1 - y) = x - 3

x = 3 + 1/(1 - y)
x = 3 - 1/(y - 1)

Then, the points of intersection (-4,4/3) and (-2,2) now become (0,4/3) and (2,2) respectively.

Do note that the line y = -x is also translated → y = -(x-4) = -x + 4


③
R can be divided into 2 parts :

-A right angled triangle from y = 2 to y = 4
-An area under the curve from y = 4/3 to y = 2

The volume of revolution obtained from the triangle, is that of a cone. It has radius of 2 units and height = (4 - 2) = 2 units.

Volume of the cone = ⅓πr²h
= ⅓π(2²)(2)
= 8/3 π units²

The volume of revolution of curve fom y = 4/3 to y = 2


= π ∫²₄ₗ₃ x² dy

= π ∫²₄ₗ₃ ( 3 - 1/(y-1) )² dy

= π ∫²₄ₗ₃ ( 9 - 6/(y - 1) + 1/(y-1)² ) dy

= π [ 9y - 6ln|y - 1| - 1/(y-1) ] ²₄ₗ₃

= π ( [9(2) - 6ln|2 - 1| - 1/(2 - 1)] - [9(4/3) - 6ln|4/3 - 1| - 1/(4/3 - 1)] )

= π ( 18 - 1 - 12 + 6 ln⅓ + 3 )

= π (8 - 6ln3)

Total volume

= 8/3 π + π(8 - 6ln3)

= π(10⅔ - 6ln3)

I will post the cylindrical shell method when I have time.

Cylindrical shell method :

①
Also involves translating the curve 4 units to the right (in the positive x-direction)

But we integrate with respect to x.

Translation :
y = x/(x + 1) → y = (x - 4)/(x - 4 + 1)

y = (x - 4)/(x - 3)
y = 1 - 1/(x - 3)


The points of intersection (-4,4/3) and (-2,2) become (0,4/3) and (2,2) respectively after translation.

The line y = - x becomes y = -(x - 4) = 4 - x


②

Volume of revolution from rotating R from x = 0 to x = 2

= (Volume from rotating y = 4 - x) - (Volume from rotating y = 1 - 1/(x - 3) )


[ Formula is 2π ∫ ph dx ,whereby :

-height of curve h (for a particular value of x) = y

p (the radius from the axis of rotation to the function, in this case it is the y-axis) = x]

= 2π ∫²₀ x(4 - x)dx - 2π ∫²₀ x(1 - 1/(x - 3)) dx

= 2π ∫²₀ (4x - x² - x + x/(x - 3) ) dx

= 2π ∫²₀ (3x - x² + 1 + 3/(x - 3) ) dx

= 2π [ 3/2 x² - ⅓x³ + x + 3ln|x - 3|] ²₀

= 2π (3/2 (2²) - ⅓(2³) + 2 + 3ln|2 - 3| - (3/2 (0²) - ⅓(0³) + 0 + 3ln|0 - 3|) )

= 2π (6 - 8/3 + 2 - 3ln3)

= 2π (5⅓ - 3ln3)

= π(10⅔ - 6ln3)