Integration

[Sweatyapples]

Find the volume generated when the area bounded by y = x^2 -1 and the x-axis is rotated about:
a) the x-axis
b) the y-axis

a) V = π ∫ {1, -1} (x^2 -1)^2 dx
= π ∫ {1, -1} (x^4 - 2x^2 + 1) dx
= π [((1/5)(1)^5 -(2/3)(1)^3 + 1) - ((1/5)(-1)^5 - (2/3)(-1)^3 -1))]
= (16π/15)u^3

b) y = x^2 -1
x^2 = y + 1
V = π ∫ {b, a} (y + 1) dy
V = π ∫ {b, a} (y + 1) dy

I'm just confused on the limits regarding (b), I've tried a number of different limits however I cannot seem to generate the correct answer (which is pie/3 according to my book).

 

[Deleted member 4993]

Find the volume generated when the area bounded by y = x^2 -1 and the x-axis is rotated about:
a) the x-axis
b) the y-axis

a) V = π∫ {1, -1} (x^2 -1)^2 dx
= π ∫ {1, -1} (x^4 - 2x^2 + 1) dx
= π [((1/5)(1)^5 -(2/3)(1)^3 + 1) - ((1/5)(-1)^5 - (2/3)(-1)^3 -1))]
= (16π/15)u^3

b) y = x^2 -1
x^2 = y + 1
V = π∫ {b, a} (y + 1) dy
V = π∫ {b, a} (y + 1) dy

I'm just confused on the limits regarding (b), I've tried a number of different limits however I cannot seem to generate the correct answer (which is pie/3 according to my book).

For these types of problems, it is always advisable to sketch the function.

What is the minimum value of y?

What is the maximum value of y?

 

[Sweatyapples]

For these types of problems, it is always advisable to sketch the function.

What is the minimum value of y?

What is the maximum value of y?

I understand that the minimum value of y = -1 and the max is ∞. I had a feeling that the limits were (0,-1), but the result of that is pie/2.

 

[stapel]

Find the volume generated when the area bounded by y = x^2 -1 and the x-axis is rotated about:
a) the x-axis
b) the y-axis

You posted this to "Intermediate / Advanced Algebra" rather than to "Calculus". What algebra formulas, methods, or algorithms have they given you for doing this without calculus? Please be complete. Thank you!