Find the volume of the resulting solid,

Question

if the region under the curve

y = 5/(x2 + 3x + 2)

from x = 0 to x = 1 is rotated about the x-axis and the y-axis.

x-axis? y-axis?

I tried solving for the rotation about the x-axis and I got

∫(from 0 to 1) 25pi / |x^2+3x+2|^2 dx and the final answer I got is 400pi(1/24 + (-ln(2)+ln(3/2))/8) but for the rotation about the y-axis I'm not sure how I can solve for that. Thank you in advance.

Answer 1

About x-axis: by disk method V = π∫₀¹25/(x² + 3x + 2)²dx = 7.171

I use TI-84. But this integral you can evaluate using partial fraction decompositiom

About y-axis: by shell method V = 2π∫₀¹5x/(x² + 3x + 2)dx = 3.7003

The same for this integral

Answer 2

To evaluate 2π ∫_{(0 to 1)} [5x / (x²+3x+2)]dx without a calculator,

5x / (x²+3x+2) = 5x / [(x+2)(x+1)] = A / (x+2) + B / (x + 1).

5x = A(x + 1) + B(x + 2)

5x = (A + B)x + (A + 2B)

So, A + B = 5 and A + 2B = 0. Therefore, A = 10 and B = -5.

The original integral is equivalent to 2π ∫_{(0 to 1)} [10/(x+2) - 5/(x+1)]dx

= 2π [10 ln(x+2) - 5 ln(x+1)]_{(0 to 1)} = 2π ([10 ln3 - 5 ln2] - [10 ln2 - 5 ln1])= 2π(10ln3 - 15ln2) ≈ 3.7