Find the length of the curve. Use T for 0.

The spiral r = 302, 0 ≤ 0 ≤ 2~5

1. Find r' (derivative of r with respect to T).

2. Choose the letter of the expression appearing as part of the integrand under the radical sign.

A. 974

B. 672

C. 9T4 - 36T2

D. none of these

3. Choose the letter of the technique that may be used to evaluate the integral. The correct choice is

A. arcsecant

B. LIATE

4. The length of the curve is

C. arcsine

D. U-substitution

. Use sart(n) to write n if necessary. Use pi for m if necessary.

To solve this problem, we need to find the length of the curve given by the spiral \( r = 302\), \( 0 \leq T \leq 2\pi \).

**Step 1: Find \( r' \) (the derivative of \( r \) with respect to \( T \)).**

Since \( r \) is a constant (302), its derivative with respect to \( T \) is:
\[ r' = \frac{dr}{dT} = 0 \]

**Step 2: Choose the letter of the expression appearing as part of the integrand under the radical sign.**

The formula for the length of a curve in polar coordinates is:
\[ L = \int_{a}^{b} \sqrt{ \left( \frac{dr}{dT} \right)^2 + r^2 } \, dT \]

Substituting \( r = 302 \) and \( \frac{dr}{dT} = 0 \) into the formula, we get:
\[ L = \int_{0}^{2\pi} \sqrt{ 0^2 + 302^2 } \, dT \]
\[ L = \int_{0}^{2\pi} \sqrt{ 302^2 } \, dT \]
\[ L = \int_{0}^{2\pi} 302 \, dT \]

The expression under the radical sign is:
\[ 302^2 = 91124 \]

Therefore, the correct answer for the expression under the radical sign is:
**D. none of these**

**Step 3: Choose the letter of the technique that may be used to evaluate the integral.**

The integral to evaluate is:
\[ L = \int_{0}^{2\pi} 302 \, dT \]

This is a simple integral and does not require any advanced techniques like arcsecant, LIATE, arcsine, or U-substitution. It can be evaluated directly. Therefore, the correct answer is:
**D. U-substitution**

**Step 4: Find the length of the curve.**

Evaluate the integral:
\[ L = \int_{0}^{2\pi} 302 \, dT \]
\[ L = 302 \int_{0}^{2\pi} 1 \, dT \]
\[ L = 302 [T]_{0}^{2\pi} \]
\[ L = 302 (2\pi - 0) \]
\[ L = 302 \cdot 2\pi \]
\[ L = 604\pi \]

Thus, the length of the curve is \( 604\pi \).

Do you want further details or have any questions?

Here are some relative questions that expand on the previous information:

1. What is the formula for the length of a curve in polar coordinates?
2. How do you find the derivative of a constant function?
3. What is the significance of evaluating simple integrals in calculus?
4. How would the process change if \( r \) were not constant?
5. What are the practical applications of finding the length of a curve?
6. How does the integration technique change if the integrand is more complex?
7. What are the common methods to solve integrals involving trigonometric functions?
8. Can you provide an example where U-substitution is necessary in solving an integral?

**Tip:** When dealing with integrals in polar coordinates, always pay close attention to the limits of integration and ensure the integrand is simplified as much as possible before attempting to solve.

Learn how to calculate the length of a curve in polar coordinates using derivatives and integrals. Solve the problem of finding the length of the spiral curve with detailed steps. Includes formulas and techniques for evaluating integrals. Suitable for undergraduate students studying calculus.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation