# Integral calculus: volume of a solid of revolution

• delapcsoncruz
In summary, the volume of the first quadrant region bounded by x=y-y3, x=1 and y=1 and revolved about the line x=1 is 121π/210 cubic units or approximately 1.81 cubic units. The solution provided used the formula dV=πR2t, where t represents the thickness and R=1-(y-y3). The final answer should be left as \frac{121}{210} \pi to follow standard math procedures.
delapcsoncruz

## Homework Statement

Find the volume of the first quadrant region bounded by x=y-y3, x=1 and y=1 that is revolved about the line x=1.

## The Attempt at a Solution

dV=∏R2t

where :

t=dy
R=1-(y-y3)
=1-y+y3

so..
dV=∏(1-y+y3)2dy
dV=∏(1-2y+y2+2y3-2y4+y6)dy
V=∏∫ from 0 to 1 of (1-2y+y2+2y3-2y4+y6)dy
V=∏(y-y2+1/3(y3)+1/2(y4)-2/5(y5)+1/7(y7) from 0 to 1

V= 121∏/210 cubic units
V= 1.81 cubic units

was my solution and final answer correct?

The setup and answer are both fine. I'm a little confused by the t, but I assume t takes the place of either dx or dy. Only thing I might suggest is to leave your answer as $\frac{121}{210} \pi$ instead of rounding, it's usually standard procedure in math courses!

ok i'll do that. you are right, 't' takes the place of either dx or dy, it is the thickness. thank you very much for your reply and time. appreciate it. thanks..

## 1. What is integral calculus?

Integral calculus is a branch of mathematics that deals with the calculation of areas, volumes, and other quantities that can be expressed as the limit of a summation. It involves finding the antiderivative of a given function and using it to calculate the area under a curve or the volume of a solid.

## 2. What is a solid of revolution?

A solid of revolution is a three-dimensional object formed by rotating a two-dimensional shape around an axis. Examples include a sphere, cylinder, and cone.

## 3. How do you find the volume of a solid of revolution using integral calculus?

To find the volume of a solid of revolution using integral calculus, you need to first determine the cross-sectional area of the shape at each point along the axis of rotation. Then, you can use the formula V = ∫ A(x) dx, where A(x) represents the cross-sectional area and dx represents an infinitely small change in the length along the axis of rotation. The integral is evaluated over the limits of the shape.

## 4. What is the difference between using the disk method and the shell method to find the volume of a solid of revolution?

The disk method involves using circular cross-sections of the shape to find the volume, while the shell method uses cylindrical shells. The disk method is typically used when the shape is revolved around a horizontal axis, while the shell method is used for shapes revolving around a vertical axis. Both methods can be used for any solid of revolution, but one may be easier or more efficient depending on the shape and axis of rotation.

## 5. Are there any real-world applications of finding the volume of a solid of revolution?

Yes, there are many real-world applications of finding the volume of a solid of revolution. For example, it can be used in engineering to calculate the volume of pipes, tanks, and other cylindrical structures. It is also used in physics to calculate the moment of inertia of objects, which is important for understanding rotational motion. In architecture and design, it can be used to calculate the volume of curved structures such as domes and arches.

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