# Find the area of the surface generated by these equations arounf the y axis

• MozAngeles
In summary: I'm having some difficulty trying to decipher some of your work. To help other readers and myself, I am fixing your LaTeX.In summary, the homework statement is trying to find the area of a surface generated by these equations around the y axis. The equations are in terms of x and y coordinates and the surface is generated by multiplying the x and y coordinates together. The attempt at a solution is to find the area of the surface generated by these equations around the y axis. The equation is in terms of x and y coordinates and the surface is generated by squaring the x and y coordinates. The equation is integrated and it yields 2t5.
MozAngeles

## Homework Statement

Find the area of the surface generated by these equations around the y axis
x=t2+1/(2t) y=4√t ; 1/√2≤t≤1

## Homework Equations

S=∫2$$\pi$$ x√(dx/dt)2+(dy/dt)2)dt

## The Attempt at a Solution

dx/dt=2t-1/2t2 squaring that, 4t2-2/t+1/(4t4)

dy/dt=2/√t squaring that , 4/t

plugging this intot he formula
S= 2$$\pi$$∫ (t2+1/(2t)√(4t2-2/t+1/(4t4)+4/t)
S= 2$$\pi$$∫ (t2+1/(2t) √((2t+1/(2t2))2
S= 2$$\pi$$∫ (t2+1/(2t) (2t+1/(2t2))

so this is where I am stuck u sub almost works here, if it weren't for that pesky negative sign. So could someone help me maybe they noticed a mistake in my work. anything thank you much.

I'm having some difficulty trying to decipher some of your work. To help other readers and myself, I am fixing your LaTeX.

Tip: If you use tex tags, use one pair for an entire equation.
MozAngeles said:

## Homework Statement

Find the area of the surface generated by these equations around the y axis
x=t2+1/(2t) y=4√t ; 1/√2≤t≤1

## Homework Equations

S=∫2$$\pi$$ x√(dx/dt)2+(dy/dt)2)dt
$$S = 2\pi \int_a^{b} x\sqrt{(dx/dt)^2+(dy/dt)^2}dt$$
MozAngeles said:

## The Attempt at a Solution

Fixed the equation below.
MozAngeles said:
dx/dt=2t-1/2t2 squaring that, 4t2-2/t+1/(4t4)

dy/dt=2/√t squaring that , 4/t

plugging this intot he formula
S= 2$$\pi$$∫ (t2+1/(2t)√(4t2-2/t+1/(4t4)+4/t)
S= 2$$\pi$$∫ (t2+1/(2t) √((2t+1/(2t2))2
S= 2$$\pi$$∫ (t2+1/(2t) (2t+1/(2t2))
$$S = 2\pi \int_{1/\sqrt{2}}^1 (t^2+1/(2t) (2t+1/(2t^2))dt$$
MozAngeles said:
so this is where I am stuck u sub almost works here, if it weren't for that pesky negative sign. So could someone help me maybe they noticed a mistake in my work. anything thank you much.

You don't need a substitution. Just multiply the two factors and then integrate. For the integrand, I get 2t3 + 5/4 + 1/(8t3)

## What is the formula for finding the area of a surface generated around the y axis?

The formula for finding the area of a surface generated around the y axis is 2π∫(f(x)*√(1+(f'(x))^2)dx, where f(x) is the equation of the curve and f'(x) is its derivative.

## Can this formula be used for any type of curve?

Yes, this formula can be used for any type of curve as long as it is rotated around the y axis. It is a generalized formula for finding the surface area of a revolution.

## What is the purpose of finding the area of a surface generated around the y axis?

The purpose of finding the area of a surface generated around the y axis is to calculate the surface area of a three-dimensional object, such as a vase or a bottle, that has a curved surface. This can be useful in fields such as engineering, architecture, and design.

## Is there a specific method for solving this type of problem?

Yes, there is a specific method for solving this type of problem called the "disk method". It involves using the formula mentioned above and integrating the function over a specified interval. This method is commonly used in calculus and is based on the concept of slicing the curved surface into infinitely thin disks.

## Are there any limitations to using this formula?

One limitation of using this formula is that it can only be used for objects that are rotated around the y axis. It cannot be used for objects rotated around other axes. Additionally, it may be challenging to apply this formula to complex curves or curves with multiple branches. In these cases, alternative methods such as the "shell method" may be used.

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