The region R is bounded by the x-axis, x = 1, x = 3, and y = 1/x^3.
Here is what I have so far:
Radius = 1/(x^3)
Area of Cross Section = pi(1/(x^3))^2
Simplified: pi(1/(x^6))
Volume = (definite integral from 1 to 3) pi(1/(x^6)) dx
= pi( -1 / 5(3)^5) - pi(-1 / 5(1)^5)
= pi (-1 / 1215) - pi (-1 / 5)
= pi(242 / 1215) = 0.625732858
Is this right?
Sincerely,
Mooch
just right. good job.
Thank you. From now on, I will include my work for reference. As you can see, I do know (to a degree) what I am doing. However, my practice instructions aren't always very clear on how to carry out these problems so I look for a second opinion to back me up. Again, thank you.
To find the volume of the region R bounded by the given curves, you first correctly identified the radius of each cross-section as 1/x^3.
The area of each cross-section is then calculated by squaring the radius and multiplying it by pi. So the area of each cross-section is pi * (1/(x^3))^2, which simplifies to pi * 1/(x^6).
To find the volume, it is indeed necessary to integrate the area of each cross-section over the interval from x = 1 to x = 3. So the volume can be obtained by evaluating the definite integral:
Volume = ∫[1 to 3] (pi * 1/(x^6)) dx
Evaluating this integral correctly gives:
Volume = [pi * (-1/5x^5)] from 1 to 3
= pi * (-1/5(3^5)) - pi * (-1/5(1^5))
= pi * (-1/1215) - pi * (-1/5)
= pi * (242/1215)
≈ 0.625732858
So your final answer of approximately 0.625732858 for the volume of region R is correct.