# Semi-circle cross section volume

• sy7ar
In summary, the conversation discusses finding the volume of a solid with a base bounded by y=\sqrt{x}, the x-axis, and x=9. The cross-sections perpendicular to the x-axis are semi-circles. The solution involves integrating (pi/8)*x from 0 to 9 and the answer is found to be 81pi/16. However, the book's answer is 81pi/8, possibly due to assuming the cross-sections are circles instead of semi-circles.
sy7ar

## Homework Statement

base of a solid is bounded by y=$\sqrt{x}$, the x-axis, and x=9. each cross-section perpendicular to the x-axis is a semi-circle. find the volume of the solid

## The Attempt at a Solution

I found the answer to be 81$\pi$/16 by the following steps:
A(semicircle)=0.5pir^2=(pi/8)*d^2 (d=diameter), and d=y=sqrt(x), so A=(pi/8)*x
integrate (pi/8)*x from 0 to 9 (pi*9^2 /16), I get the answer as 81pi/16

Last edited:
I agree with you. Maybe we are both missing something, but I can't think what it might be.

I agree with you both.

i guess the answer on my book is wrong then. I really can't think of any other ways

The book solution apparently calculated the volume assuming the cross section was a circle instead of a semi-circle.

LCKurtz said:
The book solution apparently calculated the volume assuming the cross section was a circle instead of a semi-circle.

ya, that's what I think too. The thing is, my book's never wrong before (for several times I doubted its answers and it turned out it's always correct), and this is the last question for the last unit. Hopefully it's just a decoy this time hehe.

## 1. What is a semi-circle cross section volume?

A semi-circle cross section volume is a three-dimensional shape that is formed when a semi-circle (half of a circle) is rotated around a central axis, creating a solid object with a circular base and a flat face.

## 2. How is the volume of a semi-circle cross section calculated?

The volume of a semi-circle cross section can be calculated by using the formula V = (1/2)πr^2h, where r is the radius of the semi-circle and h is the height of the solid object.

## 3. What are some real-life applications of semi-circle cross section volume?

Semi-circle cross section volumes can be found in various objects such as pipes, tunnels, and containers. They are also commonly used in architecture and engineering for creating curved structures and domes.

## 4. How does the volume of a semi-circle cross section compare to that of a full circle?

The volume of a semi-circle cross section is half of the volume of a full circle with the same radius. This is because a semi-circle only covers half of the circle's circumference, resulting in a smaller volume.

## 5. Can the volume of a semi-circle cross section be calculated for irregular shapes?

Yes, the volume of a semi-circle cross section can be calculated for irregular shapes as long as the base is a semi-circle. The formula V = (1/2)πr^2h can still be used, but the radius and height may need to be measured or estimated differently for each irregular shape.

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