Volume of revolution in the first quadrant?

In summary: In that formula, ##y_{upper}## is the rightmost curve, and ##y_{lower}## is the leftmost curve. In this problem, you want to use ##y_{upper} = 3x+2## and ##y_{lower} = x^3##. So you need to find the limits of integration for the region that you want to use. So, the rightmost curve is ##y = 3x +2## (or ##x = \frac{y-2}{3}##), and the leftmost curve is ##y = x^3## (or ##x = \sqrt[3]{y}##). So the limits of integration are at ##x = 0
  • #1
CAH
48
0

Homework Statement


Find the volumes of the solid formed when each of the areas in the following perform one revolution about the X axis...
Question: The volume line in the first quadrant and bounded by the curve y=x^3 and the line y=3x+2.

Homework Equations


Volume of revolution about X-axis: V=pi*integral(y^2) dx. ('b' upper, and 'a' lower limit)

The Attempt at a Solution


Ok so I can find the volume (I think) of the whole system, but I can't just find the volume in the first quadrant.
V=pi*integral[(3x+2)^2 - (x^3)^2] between 2 and -1. So this is area under line minus area under curve, between there intersections. I don't know how to find the area in the first quadrant alone though!
I got V= 264pi/7 (the whole system) and the answer in my textbook is 56pi/5. Also sometimes the textbooks are wrong.
Thanks
 
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  • #2
CAH said:

Homework Statement


Find the volumes of the solid formed when each of the areas in the following perform one revolution about the X axis...
Question: The volume line in the first quadrant and bounded by the curve y=x^3 and the line y=3x+2.

Homework Equations


Volume of revolution about X-axis: V=pi*integral(y^2) dx. ('b' upper, and 'a' lower limit)

The Attempt at a Solution


Ok so I can find the volume (I think) of the whole system, but I can't just find the volume in the first quadrant.
V=pi*integral[(3x+2)^2 - (x^3)^2] between 2 and -1.
In the first quadrant, x > 0, so your integral should not be from - 1 to 2. The region that is being revolved is bounded above by the line y = 3x + 2, below by the curve y = x3, and on the left by the y-axis (because you're interested only in what's happening in Quadrant 1).
CAH said:
So this is area under line minus area under curve, between there intersections. I don't know how to find the area in the first quadrant alone though!
I got V= 264pi/7 (the whole system) and the answer in my textbook is 56pi/5. Also sometimes the textbooks are wrong.
Thanks
 
  • #3
Yea i did the 0-2 boundary, must have accidentally wrote -1! But I still have the area beneath the X axis in my equation and I don't know how to take it away so it's only the area in the first quad left...?
 
  • #4
CAH said:

Homework Statement


Find the volumes of the solid formed when each of the areas in the following perform one revolution about the X axis...
Question: The volume line in the first quadrant and bounded by the curve y=x^3 and the line y=3x+2.

Homework Equations


Volume of revolution about X-axis: V=pi*integral(y^2) dx. ('b' upper, and 'a' lower limit)

Remember, that formula is for the area between the ##x## axis and the function, rotated. You need the formula$$
A =\pi \int_a^b y_{upper}^2 - y_{lower}^2~dx$$to get the volume of the area between the curves rotated.
 

FAQ: Volume of revolution in the first quadrant?

1. What is the definition of volume of revolution in the first quadrant?

The volume of revolution in the first quadrant is a mathematical concept that involves finding the volume of a three-dimensional object formed by rotating a two-dimensional shape around a specified axis within the first quadrant of a coordinate plane.

2. What is the formula for calculating volume of revolution in the first quadrant?

The formula for calculating volume of revolution in the first quadrant is V = π∫ab (f(x))2 dx, where a and b are the limits of integration and f(x) is the function representing the curve being rotated.

3. What is the difference between volume of revolution in the first quadrant and volume of revolution in general?

The main difference between volume of revolution in the first quadrant and volume of revolution in general is the restriction of the area being rotated to the first quadrant in the former. This means that the resulting volume will only be in the positive x and y directions.

4. How do you approach finding the volume of revolution in the first quadrant?

To find the volume of revolution in the first quadrant, you first need to identify the function representing the curve being rotated and the limits of integration. Then, you can use the formula V = π∫ab (f(x))2 dx to calculate the volume.

5. What are some real-life applications of volume of revolution in the first quadrant?

Volume of revolution in the first quadrant has many practical applications in fields such as engineering, architecture, and physics. For example, it can be used to calculate the volume of a water tank, the capacity of a cylindrical container, or the mass of a solid object with a curved surface.

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