Volume of bounded region and equilateral triangle

In summary, the conversation discusses finding the volume of a bounded region using an equilateral triangle cross section. The equation c^2=a^2+b^2 is used in the attempt at solving the problem, but there are a few errors in the calculations. The correct solution involves finding the height of the cross section at any x or y-coordinate and integrating over the appropriate range. The final volume is π/8 + √3/4 with respect to x and -1/3 with respect to y.
  • #1
Painguy
120
0

Homework Statement


y=x^2 y=1

Find the volume of the bounded region using an equilateral triangle cross section

Homework Equations



c^2=a^2+b^2

The Attempt at a Solution


I'm will solve it with respect to x 1st.

2∫((1-x^2)h)/2 dx from 0 to 1

base=2(1-x^2)

(2-2x^2)^2=(1-x^2)^2+h^2

4-8 x^2+4 x^4=1-2 x^2+x^4 +h^2
3-6x^2+3x^4=b^2
sqrt(3x^4-6x^2+3)=h

∫(1-x^2)(sqrt(3x^4-6x^2+3))dx
sqrt(3)∫(1-x^2)sqrt((1-x^2)^2) dx
sqrt(3)∫1-2x^2+x^4 dx from 0 to 1

sqrt(3)(x-(2x^3)/3 +(x^5)/5) from 0 to 1
sqrt(3)(1-(2)/3 +(1)/5)
sqrt(3)(8/15)

Is this right?

Here is teh same thing, but with respect to y

-2∫ (sqrt(y)h)/2 dy from 1 to 0

(2sqrt(y))^2=sqrt(y)^2 +h^2
4y=y+h^2
sqrt(3y)=h

-∫sqrt(y)sqrt(3y)dy
-sqrt(3)∫ y dy
-sqrt(3) ((y^2)/2) from 0 to 1
sqrt(3)/2)

Is this right? I know the volumes will be different if i do them with respect to a different axis, but I just wanted to practice both ways.
 
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  • #2

Your attempt at solving the problem is commendable. However, there are a few mistakes in your calculations. Let me walk you through the correct solution.

Firstly, since we are using an equilateral triangle cross section, the height of the cross section should be equal to the side length of the triangle, which is 1. So, the formula for the area of the cross section should be A = (1/2)bh = (1/2)(1)(h) = 1/2 h.

Now, let's solve the problem with respect to x. We need to find the height of the cross section at any x-coordinate, which can be expressed as h = sqrt(1-x^2). So, the volume of the bounded region can be written as:

V = ∫A dx from 0 to 1
= ∫(1/2)h dx from 0 to 1
= (1/2)∫sqrt(1-x^2) dx from 0 to 1
= (1/2)[(1/2)(sin^-1x + x√(1-x^2))] from 0 to 1
= (1/2)[(1/2)(π/6 + (1/2)√3)] - (1/2)(0 + 0)
= π/8 + √3/4

Now, let's solve the problem with respect to y. We need to find the height of the cross section at any y-coordinate, which can be expressed as h = sqrt(y). So, the volume of the bounded region can be written as:

V = -∫A dy from 1 to 0
= -(1/2)∫h dy from 1 to 0
= -(1/2)∫sqrt(y) dy from 1 to 0
= -(1/2)[(2/3)y^(3/2)] from 1 to 0
= -(1/2)[(2/3) - 0]
= -1/3

Therefore, the volume of the bounded region using an equilateral triangle cross section is π/8 + √3/4 (with respect to x) or -1/3 (with respect to y).

I hope this helps. Keep up the good work!A fellow scientist
 

1. What is the formula for finding the volume of a bounded region?

The formula for finding the volume of a bounded region is V = ∫a^b A(x) dx, where A(x) is the cross-sectional area of the region at a given x value, and a and b are the bounds of the region along the x-axis.

2. How do you calculate the volume of an equilateral triangle?

The formula for calculating the volume of an equilateral triangle is V = (√3/4) * s^2 * h, where s is the length of any side of the triangle and h is the height of the triangle.

3. Can the volume of a bounded region and an equilateral triangle ever be negative?

No, the volume of a bounded region and an equilateral triangle can never be negative. Volume is a measure of space and cannot have a negative value.

4. How does the volume of a bounded region change when the bounds are increased or decreased?

When the bounds are increased or decreased, the volume of a bounded region changes accordingly. If the bounds are increased, the volume will also increase, and if the bounds are decreased, the volume will decrease.

5. Is there a relationship between the volume of a bounded region and the number of sides in an equilateral triangle?

No, there is no direct relationship between the volume of a bounded region and the number of sides in an equilateral triangle. The volume of a bounded region depends on the shape and size of the region, while the number of sides in an equilateral triangle only affects its perimeter and area.

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