1:Find the volume of the solid in the 1st octant bounded by the coordinate planes,the plane x=3 ,& the parabolic cylinder z= 4-y^2.2 & 3 questions...2: If r with arrow on the top (t)=...

1:Find the volume of the solid in the 1st octant bounded by the coordinate planes,the plane x=3 ,& the parabolic cylinder z= 4-y^2.

2 & 3 questions...

2: If r with arrow on the top (t)= ln(t^2+1)i with ^ on the top +(Tan^-1 t) j with ^ on the top + (t^2 + 1)^1/2 K with ^ on the top is the position of a particle in space at time "t" .Find the angle between the velocity and acceleration vectors at time t=0

3: For the complete revolution( 0≤ t≤ 2π ) of the helix curve: x = Cost, y = Sint and z = t, evaluate . integral with lower limit C y.Sin z ds.

Expert Answers
sciencesolve eNotes educator| Certified Educator

You need to find the angle between the velocity vector and the acceleration vector, hence, you need to evaluate the following dot product, such that:

`bar v(t)* bar a(t) = |bar v(t)|*|bar a(t)|*cos (hat(bar v(t),bar a(t)))`

You need to determine the velocity vector, hence, you need to differentiate the position vector, with respect to t, such that:

`(bar r(t))' = ((ln (t^2 + 1)) bar i + (tan^(-1) t) bar j + sqrt(t^2 + 1) bar k)'`

`bar v(t) = (bar r(t))' = (2t)/(t^2+1) bar i + 1/(1 + t^2) bar j + t/(sqrt(t^2+1)) bar k`

You need to determine the acceleration vector, hence, you need to differentiate the velocity vector, with respect to t, such that:

`bar a(t) = ((2t)'(t^2+1) - 2t(t^2+1)')/((t^2+1)^2) bar i - (2t)/((1 + t^2)^2) bar j + (t'sqrt(t^2+1) - (t^2)/(sqrt(t^2+1)))/(t^2+1) bar k`

`bar a(t) = (2t^2 + 2 - 4t^2)/((t^2+1)^2) bar i - (2t)/((1 + t^2)^2) bar j + (sqrt(t^2+1) - (t^2)/(sqrt(t^2+1)))/(t^2+1) bar k`

You need to evaluate `bar v(t)*bar a(t)` such that:

`bar v(t)*bar a(t) = (2t(2 - 2t^2))/((t^2+1)^3) - (2t)/((t^2+1)^3) + t^3/((t^2+1)^3)`

`bar v(t)*bar a(t) = (4t - 4t^3 - 2t + t^3)/((t^2+1)^3)`

`bar v(t)*bar a(t) = (2t - 3t^3)/((t^2+1)^3)`

You need to evaluate `|bar v(t)|` such that:

`|bar v(t)| = sqrt((4t^2)/((t^2+1)^2) + 1/((t^2+1)^2) + t^2/(t^2+1))`

`|bar v(t)| = sqrt((4t^2 + 1 + t^4 + t^2)/((t^2+1)^2))`

`|bar v(t)| = (sqrt(t^4 + 5t^2 + 1))/(t^2+1)`

`|bar a(t)| = sqrt(((2 - 2t^2)^2)/((t^2+1)^4) + (4t^2)/((1 + t^2)^4) + t^4/((t^2+1)^2))`

`|bar a(t)| = (sqrt (4 - 8t^2 + 4t^4 + 4t^2 + t^8 + 2t^6 +t^4))/((1 + t^2)^2)`

`|bar a(t)| = (sqrt (t^8 + 2t^6 + 5t^4 - 4t^2 + 4))/((1 + t^2)^2)`

`cos (hat(bar v(t),bar a(t))) = (|bar v(t)|*|bar a(t)|)/(bar v(t)*bar a(t))`

Hence, evaluating `cos (hat(bar v(t),bar a(t)))` yields `cos (hat(bar v(t),bar a(t))) = (|bar v(t)|*|bar a(t)|)/(bar v(t)*bar a(t)).`