HumanBein
Answered

Explore Westonci.ca, the leading Q&A site where experts provide accurate and helpful answers to all your questions. Explore in-depth answers to your questions from a knowledgeable community of experts across different fields. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.

Calculus again!

A yo-yo is moving up and down a string so that its velocity at time t is given by v(t) = 4cos(t) for time t ≥ 0. The initial position of the yo-yo at time t = 0 is x = 4. Find the displacement and distance from time t = 0 to time t = π.
I know that the displacement would be calculated by [tex]\int\limits^0_\pi {4cos(t)} \, dx[/tex] and the displacement would be calculated by [tex]\int\limits^0_\pi {|4cos(t)|} \, dx[/tex], but would both be 0? Thank you!

Sagot :

jaimitoM

Not. Both are diferent.

The displacement  from time t = 0 to time t = π is given by:

[tex]{\displaystyle s = \int _0^{\pi }4\cos\left(t\right)\:dt}[/tex]

[tex]{\displaystyle s = 4\left[\sin \left(t\right)\right]^{\pi }_0}[/tex]

[tex]s = 4[0 - 0 ][/tex]

[tex]s = 0[/tex]

The distance from time t = 0 to time t = π is:

[tex]{\displaystyle d = \int _0^{\pi }|4\cos\left(t\right)|\:dt}[/tex]

[tex]{\displaystyle d = \int _0^{\frac{\pi }{2}}4\cos \left(t\right)dt+\int _{\frac{\pi }{2}}^{\pi }-4\cos \left(t\right)dt}[/tex]

[tex]d = 4\left[\sin \left(t\right)\right]^{\frac{\pi}{2}}_0 - 4\left[\sin \left(t\right)\right]^{\pi}_{\frac{\pi}{2}}[/tex]

[tex]d = 4 + 4[/tex]

[tex]d = 8[/tex]

We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Thank you for visiting Westonci.ca, your go-to source for reliable answers. Come back soon for more expert insights.