Given
The parametric curve is defined as:
\[ x = \cos\theta,\quad y = \sin\theta,\quad z = \theta \]
The parameter \(\theta\) varies from:
\[ \theta = 0 \quad \text{to} \quad \theta = 2\pi \]
Formula for Arc Length of a Space Curve
For a space curve defined parametrically by:
\[ x = x(t),\; y = y(t),\; z = z(t) \]
the arc length \(L\) from \(t=a\) to \(t=b\) is given by:
\[ L = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2 + \left(\frac{dz}{dt}\right)^2}\, dt \]
Step 1: Compute the Derivatives
\[ \frac{dx}{d\theta} = -\sin\theta \]
\[ \frac{dy}{d\theta} = \cos\theta \]
\[ \frac{dz}{d\theta} = 1 \]
Step 2: Substitute into the Arc Length Formula
\[ \left(\frac{dx}{d\theta}\right)^2 + \left(\frac{dy}{d\theta}\right)^2 + \left(\frac{dz}{d\theta}\right)^2 \]
\[ = (-\sin\theta)^2 + (\cos\theta)^2 + 1^2 \]
\[ = \sin^2\theta + \cos^2\theta + 1 \]
\[ = 1 + 1 = 2 \]
Step 3: Evaluate the Integral
\[ L = \int_0^{2\pi} \sqrt{2}\, d\theta \]
\[ = \sqrt{2} \int_0^{2\pi} d\theta \]
\[ = \sqrt{2}\,(2\pi) \]
\[ = 2\pi\sqrt{2} \]
Step 4: Numerical Approximation
\[ 2\pi\sqrt{2} \approx 2 \times 3.1416 \times 1.4142 \]
\[ \approx 8.8858 \]
Rounded off to one decimal place:
\[ \boxed{8.9} \]
Final Answer
The arc length of the given parametric curve is:
\[ \boxed{8.9} \]
Important Questions from Application Of Definite Integral (Area)
The equation of a closed curve in two-dimensional polar coordinates is given by $r = \frac{2}{\sqrt{\pi}}(1 - \sin \theta)$. The area enclosed by the curve is ______ (answer in integer).
The area bounded by the curves, $y = \sqrt{x}$, and $y = 8x^2$ is _______________(rounded off to 3 decimal places)
Two straight lines pass through the origin $(x_0, y_0) = (0,0)$. One of them passes through the point $(x_1, y_1) = (1,3)$ and the other passes through the point $(x_2, y_2) = (1,2)$.
What is the area enclosed between the straight lines in the interval $[0, 1]$ on the x-axis?
Consider the equation for a curve, $y = f(x) = x^2 + x$.
The area enclosed by the curve, the x -axis ($y = 0$ line); the vertical lines passing through $x = 1$ and $x = 2$ is _________ (rounded off to 2 decimal places)- The area of the region (rounded off to one decimal place) enclosed between the curves $y = x$ and $y = 3\sqrt{x}$ and between the lines $x = 0$ and $x = 1$ is ________ units.
