https://file.aiquickdraw.com/mathbot/file/17270298738087sk7ln8n.jpg

To solve this problem, we need to find the arc length of a parametric curve defined by:

\[
x(t) = \sin^2(t), \quad y(t) = \cos^2(t), \quad 0 \leq t \leq \frac{\pi}{2}.
\]

The formula for the arc length \( L \) of a parametric curve is given by:

\[
L = \int_{t_1}^{t_2} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt.
\]

### Step 1: Compute \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\)

First, differentiate \(x(t)\) and \(y(t)\) with respect to \(t\):

\[
\frac{dx}{dt} = \frac{d}{dt}(\sin^2(t)) = 2\sin(t)\cos(t) = \sin(2t).
\]

\[
\frac{dy}{dt} = \frac{d}{dt}(\cos^2(t)) = -2\cos(t)\sin(t) = -\sin(2t).
\]

### Step 2: Apply the arc length formula

Substitute these derivatives into the arc length formula:

\[
L = \int_0^{\frac{\pi}{2}} \sqrt{(\sin(2t))^2 + (-\sin(2t))^2} \, dt.
\]

Since both \((\sin(2t))^2\) and \((- \sin(2t))^2\) give the same result:

\[
L = \int_0^{\frac{\pi}{2}} \sqrt{2\sin^2(2t)} \, dt.
\]

This simplifies further as:

\[
L = \int_0^{\frac{\pi}{2}} \sqrt{2} \cdot |\sin(2t)| \, dt.
\]

Since \(\sin(2t)\) is non-negative on \([0, \frac{\pi}{2}]\), the absolute value can be removed:

\[
L = \sqrt{2} \int_0^{\frac{\pi}{2}} \sin(2t) \, dt.
\]

### Step 3: Evaluate the integral

The integral of \(\sin(2t)\) is:

\[
\int \sin(2t) \, dt = -\frac{1}{2}\cos(2t).
\]

Thus, the arc length is:

\[
L = \sqrt{2} \left[ -\frac{1}{2} \cos(2t) \right]_0^{\frac{\pi}{2}}.
\]

Evaluate at the bounds:

At \(t = \frac{\pi}{2}\):

\[
\cos(2 \cdot \frac{\pi}{2}) = \cos(\pi) = -1.
\]

At \(t = 0\):

\[
\cos(0) = 1.
\]

Thus:

\[
L = \sqrt{2} \left( -\frac{1}{2}(-1) + \frac{1}{2}(1) \right) = \sqrt{2} \cdot \frac{1}{2} \cdot 2 = \sqrt{2}.
\]

### Conclusion:

The correct integral that gives the arc length of the parametric curve is:

\[
\boxed{C} \, \int_0^{\frac{\pi}{2}} \sqrt{\sin^4(t) + \cos^4(t)} \, dt.
\]

Would you like further details on any of the steps? Here are some related questions:

1. How does parametric differentiation work?
2. Can you explain more about calculating the arc length for parametric curves?
3. What other methods are there to calculate arc length?
4. Why is the formula for arc length based on the Pythagorean theorem?
5. What happens if the curve is not differentiable?

**Tip**: To avoid errors in complex parametric problems, always verify the range and behavior of trigonometric functions within the given interval.

Which of the following gives the length of the curve defined by the parametric equations x(t) = sin^2(t) and y(t) = cos^2(t) for 0 <= t <= π/2?

This problem explores how to find the arc length of the parametric curve x(t) = sin^2(t) and y(t) = cos^2(t) within the interval 0 <= t <= π/2. The solution involves calculating derivatives and applying the arc length formula.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation