1)Find the arc length 𝑠 of the vector function 𝐫(𝑡)=3cos3(𝑡)𝐢+3sin3(𝑡)𝐣+𝐤 from 𝑡=0 to t=π/2. 【s = ?】 2)Find the arc leng

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1)Find the arc length 𝑠 of the vector function 𝐫(𝑡)=3cos3(𝑡)𝐢+3sin3(𝑡)𝐣+𝐤 from 𝑡=0 to t=π/2. 【s = ?】
2)Find the arc length 𝑠 of the vector function 2𝑡𝐢+𝑡𝐣+3𝑡𝐤 from 𝑡=2 to 𝑡=4.【s = ?】
(Give an exact answer. Use symbolic notation and fractions where needed.)
3)A projectile is fired at an angle of 30∘ to the horizontal with an initial speed of 560 m/s. What are its range, the time of flight, and the greatest height reached?
【range ≈ ?m】【time of flight ≈ ?s】【maximum height ≈ ?m】
(Use decimal notation. Give your answers as whole numbers. Assume 𝑔=9.8 m/s2.)
4)Solve the vector differential equation with the condition 𝐫(0)=𝐢+𝐤.
𝐫′(𝑡)=cos(5𝑡)𝐢+sin(5𝑡)𝐣+5𝐤 【𝐫(𝑡)=？】
5)Solve the vector differential equation with the condition 𝐫(0)=𝐢−𝐣.
𝐫′(𝑡)=7sin(𝑡)𝐢+cos(𝑡)𝐣+𝐤 【𝐫(𝑡)=？】
(Give your answer using component form or standard basis vectors. Express numbers in exact form. Use symbolic notation and fractions where needed.)