2x 1 Suppose you wish to apply the FTOLI to the line integral + ( — — — )^-*-^\ (1) - 1/2 ^ where C is the curve parametrized by F(1) = 1²î +21ĵ + (1³+1) for 1

Differentiate: 1. y''' - 4y'' + y' + 6y = 0 2. yIV + 16y = 0

Find the length of the curve r(t) = 6t^3i − 2t^3j − 3t^3 k from t = 1 to t= 2. Image is also attached.

Q.3) (a) Evalute |(22 – y?)dx + 2ydy, along the circular arc C that is upper half circle oriented counterclock wise. (b) Can you apply the Green's theoren to evalute the line integral in the above question. Explain your ideas.

The plane curve C is the part of y = x^4 that begins at the point (−1, 1) and ends at the point(1, 1). Evaluate the line integral

Evaluate the line integral (8x + 7y)dx + (2x - 7y)dy along the | curve C: r = 8 cos t, y = 16 sin t (0 < ts-) T NOTE: Enter the exact answer. | (8x + 7y)dx + (2x – 7y)dy =

Find the orthogonal trajectories of the family of curves: y = k (csc x + cotx) Ⓒy² = 2 (C- cos x) O y = C-2sec x x= 2y (C-sinx) Ox2=2 (C-siny)

Find V. (V × F), if F(x, y, z) = 3e"²i+ 4xe'j – 2e²k. V·(V × F)

Find the I.F of 2y(x^2-y+x)dx+(x^2-2y)dy=0