Asksia AI LOGO

Sia

Question
Math
Posted 2 months ago

Find the length of the curve.
x=t22,y=(2t+1)3/23,0t10
x=\frac{t^{2}}{2}, y=\frac{(2 t+1)^{3 / 2}}{3}, 0 \leq t \leq 10


The length of the curve is \square (Simplify your answer.)
Sign in to unlock the answer
Answer from Sia
Posted 2 months ago
Solution by Steps
step 1
To find the length of the curve, we use the formula for the arc length of a parametric curve: L=ab(dxdt)2+(dydt)2dt L = \int_{a}^{b} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt where x=t22 x = \frac{t^2}{2} and y=(2t+1)3/23 y = \frac{(2t+1)^{3/2}}{3} , and 0t10 0 \leq t \leq 10
step 2
First, we find dxdt\frac{dx}{dt}: dxdt=ddt(t22)=t \frac{dx}{dt} = \frac{d}{dt} \left( \frac{t^2}{2} \right) = t
step 3
Next, we find dydt\frac{dy}{dt}: dydt=ddt((2t+1)3/23)=32(2t+1)1/223=(2t+1)1/2 \frac{dy}{dt} = \frac{d}{dt} \left( \frac{(2t+1)^{3/2}}{3} \right) = \frac{3}{2} \cdot \frac{(2t+1)^{1/2} \cdot 2}{3} = (2t+1)^{1/2}
step 4
Substitute dxdt\frac{dx}{dt} and dydt\frac{dy}{dt} into the arc length formula: L=010t2+(2t+1)dt L = \int_{0}^{10} \sqrt{t^2 + (2t+1)} \, dt L=010t2+2t+1dt L = \int_{0}^{10} \sqrt{t^2 + 2t + 1} \, dt L=010(t+1)2dt L = \int_{0}^{10} \sqrt{(t+1)^2} \, dt L=010t+1dt L = \int_{0}^{10} |t+1| \, dt
step 5
Since t t ranges from 0 to 10, t+1 t+1 is always positive, so t+1=t+1|t+1| = t+1: L=010(t+1)dt L = \int_{0}^{10} (t+1) \, dt L=[t22+t]010 L = \left[ \frac{t^2}{2} + t \right]_{0}^{10} L=(1022+10)(022+0) L = \left( \frac{10^2}{2} + 10 \right) - \left( \frac{0^2}{2} + 0 \right) L=(50+10)0 L = \left( 50 + 10 \right) - 0 L=60 L = 60
Answer
The length of the curve is 60.
Key Concept
Arc Length of Parametric Curve
Explanation
To find the length of a curve given by parametric equations, we use the arc length formula, which involves integrating the square root of the sum of the squares of the derivatives of the parametric equations.

Not the question you are looking for? Ask here!

Enter question by text

Enter question by image

Upgrade to Asksia Pro

Join a AskSia's Pro Plan, and get 24/7 AI tutoring for your reviews, assignments, quizzes and exam preps.

Unlimited chat query usages
Strong algorithms that better know you
Early access to new release features
Study Other Question