4. (a) Evaluate: ſc (ry*) ds where C is the right half circle, r² + y² = 25 traced out in a counter clockwise direction. [Hints: ds = V[r'(t))² + [v/(t)]*dt] (b) Evaluate e dx where C is the portion of x = sin (4y) from y = to y = 7.

Q: 3. [32] −1 4-3 4. 5. [. |_3 -4 2 1 2 -1 4,

A:

Q: (3/5) What is the directional derivative Duf at the point (3, 1)? Let f (x, y) = x² - y and u =

A:

Q: Vỹ - 2 dy (b). / ž r64 7. Evaluate (a). tan(4x + 9) dx

A: Given: To evaluate a) ∫164y-2y3dy                     b) ∫13tan(4x+9) dx

Q: SOLVE BY LINEAR METHOD - y'- 3 X Y = x 3 cos (x)

A:

Q: Let dy Then = dx y = tan -¹(√42²-1)

A:

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.