Show that the length of the arc of the curve 9y2=x(3-x) the whole square measured from the origin to pt x,y 3r2=3y2+4x2
To find the length of the arc of the curve, we need to use the arc length formula:
L = ∫√(1 + (dy/dx)²) dx
First, we need to find dy/dx by implicitly differentiating the given equation:
9y² = x(3-x)²
18y(dy/dx) = (3-x)² - 2x(3-x)(-1)
18y(dy/dx) = (3-x)² + 2x(3-x)
dy/dx = [(3-x)² + 2x(3-x)] / (18y)
Simplifying further:
dy/dx = [3(x-1)(x-5)] / (18y)
dy/dx = (x-1)(x-5) / (6y)
Now, we can substitute this expression into the arc length formula:
L = ∫√(1 + [(x-1)(x-5) / (6y)]²) dx
To make the integral easier to evaluate, we can use the equation of the curve to eliminate y:
9y² = x(3-x)²
y² = x(3-x)² / 9
y = ±(x(3-x) / 3)^(1/2)
We only need to consider the positive square root since we are measuring the arc length from the origin. Substituting for y:
L = ∫√(1 + [(x-1)(x-5) / (6(x(3-x) / 3)^(1/2))]²) dx
Simplifying:
L = ∫√(1 + [(x-1)(x-5) / (2(x(3-x))^(1/2))]²) dx
L = ∫√(1 + [(x-1)(x-5) / (2x(3-x))]²) dx
Now, we need to find the limits of integration. Since we are measuring the arc length from the origin, we need to find the x-coordinate of the point (3r²/(3+y²), y) on the curve. Substituting for y and simplifying:
3r²/(3+(x(3-x)/3))² = 3y²/(3+y²)
9r^4 = 3(3+y²)^2 x (3-x)²
y² = [(r²(3-x))^(2/3) - 3³/²] / 3
y = [(r²(3-x))^(1/3) - 3^(1/2)] / (3^(1/2))^(1/3)
We can use this expression as the upper limit of integration. The lower limit is simply 0, since we are measuring the arc length from the origin. Therefore:
L = ∫₀^[(r²(3-x))^(1/3) - 3^(1/2)] / (3^(1/2))^(1/3) √(1 + [(x-1)(x-5) / (2x(3-x))]²) dx
This integral is difficult to evaluate analytically. We can use numerical methods or approximations to find an estimate of the arc length.
You made a mistake right at the start. It should have been
9y² = x(3-x)²
18y(dy/dx) = (3-x)² + 2x(3-x)(-1)
Simplifying things a bit, we can start off with
y = 1/3 √x (x-3)
y' = (x-1)/(2√x)
L = ∫√(1 + y'^2) dx = ∫√(1 + (x-1)^2/(4x)) dx
= ∫(x+1)/(2√x) dx = 1/3 √x (x+3)
Not sure what that r^2=3y^2+4x^2 has to do with it. We need the x-coordinate of the point; you have calculated the y-coordinate.
You are correct, I apologize for the mistake in my initial calculation. Thank you for pointing that out. Your derivation of the arc length formula is correct and gives the answer as:
L = 1/3 √x (x+3)
Regarding the equation r^2 = 3y^2 + 4x^2, you are also correct that it was unnecessary for this problem since we only needed to find the x-coordinate of the given point. I apologize for including irrelevant information. Thank you for catching that error as well.
To find the length of the arc of a curve between two points, we need to use the concept of arc length integration.
For the first curve, the equation is 9y^2 = x(3-x)^2.
1. To begin, let's find the derivative of y with respect to x. We differentiate both sides of the equation:
d/dx (9y^2) = d/dx (x(3-x)^2)
18yy' = (3-x)^2 + x * 2(3-x)(-1)
2. Simplifying this equation, we get:
18yy' = (3-x)^2 - 2x(3-x)
3. Now, let's find the expression for y' (the derivative of y with respect to x):
y' = [(3-x)^2 - 2x(3-x)] / 18y
4. The arc length is given by the formula:
L = ∫[a, b] √(1 + (y')^2) dx
where [a, b] represents the interval of x-values we want to find the arc length, and √ represents the square root.
5. Substituting the expression for y', the arc length integral becomes:
L = ∫[a, b] √[1 + {[(3-x)^2 - 2x(3-x)] / 18y}^2] dx
6. By substituting the given values for x and y, we can calculate the arc length from the origin to the point (x, y).
For the second curve, the equation is 3r^2 = 3y^2 + 4x^2.
1. Since this equation is in terms of polar coordinates, we need to convert it to Cartesian coordinates before proceeding with finding the arc length.
2. Converting the equation to Cartesian coordinates, we get:
3(x^2 + y^2) = 3y^2 + 4x^2
3x^2 + 3y^2 = 3y^2 + 4x^2
x^2 = y^2
3. We can rewrite this equation as:
x^2 - y^2 = 0
4. Rearranging further, we have:
(x - y)(x + y) = 0
5. From this rearranged equation, we can see that the curve consists of two lines: x - y = 0 and x + y = 0.
The length of each line segment can be calculated using the distance formula or by taking the absolute values of the intercepts of these lines on the axes.
Therefore, the length of the arc of the curve 3r^2 = 3y^2 + 4x^2 from the origin to the point (x, y) would depend on the specific values of x and y.