Multiple Choice
Find the limit:
100
views
1. Limits and Continuity
Finding Limits Algebraically
Multiple Choice
Find the exact length of the curve given by , for .
A
B
C
D
Verified step by step guidance1
Step 1: Recall the formula for the arc length of a parametric curve: \( L = \int_{a}^{b} \sqrt{\left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2} \, dt \). Here, \( x \) and \( y \) are given as functions of \( t \), and \( a \) and \( b \) are the bounds of \( t \).
Step 2: Compute \( \frac{dx}{dt} \) and \( \frac{dy}{dt} \). For \( x = 2 \sqrt[3]{t^3} \), differentiate with respect to \( t \) to get \( \frac{dx}{dt} = 2 \cdot \frac{d}{dt}(t) = 2 \). For \( y = t^2 - 2 \), differentiate with respect to \( t \) to get \( \frac{dy}{dt} = 2t \).
Step 3: Substitute \( \frac{dx}{dt} \) and \( \frac{dy}{dt} \) into the arc length formula. This gives \( L = \int_{0}^{4} \sqrt{(2)^2 + (2t)^2} \, dt \). Simplify the expression inside the square root: \( \sqrt{4 + 4t^2} \).
Step 4: Factor out the constant from the square root to simplify further: \( \sqrt{4(1 + t^2)} = 2\sqrt{1 + t^2} \). The arc length formula now becomes \( L = \int_{0}^{4} 2\sqrt{1 + t^2} \, dt \).
Step 5: Evaluate the integral \( \int_{0}^{4} 2\sqrt{1 + t^2} \, dt \) using appropriate techniques, such as substitution or recognizing it as a standard integral form. This will yield the exact length of the curve.
Related Videos
Related Practice
