Question

The value of the integral

∫C(3x2+ycosx)dx+(sinx−4y3)dy∫C(3x2+ycosx)dx+(sinx−4y3)dy,

where CC is an arbitrary path from A(−π,−1)A(−π,−1) to B(2π,1)B(2π,1), is:

Answer #1

Problem 7. Consider the line integral Z C y sin x dx − cos x
dy.
a. Evaluate the line integral, assuming C is the line segment
from (0, 1) to (π, −1).
b. Show that the vector field F = <y sin x, − cos x> is
conservative, and find a potential function V (x, y).
c. Evaluate the line integral where C is any path from (π, −1)
to (0, 1).

Evaluate Integral (subscript c) z dx + y dy − x dz, where the
curve C is given by c(t) = t i + sin t j + cost k for 0 ≤ t ≤
π.

Solve the initial value problems.
1) x (dy/dx)+ 2y = −sinx/x , y(π) = 0.
2) 3y”−y=0 , y(0)=0,y’(0)=1.Use the power series method
for this one . And then solve it using the characteristic
method
Note that 3y” refers to it being second order
differential and y’ first

a). Find dy/dx for the following integral.
y=Integral from 0 to cosine(x) dt/√1+ t^2 ,
0<x<pi
b). Find dy/dx for tthe following integral
y=Integral from 0 to sine^-1 (x) cosine t dt

Evaluate the line integral ∫F⋅d
r∫CF⋅d r where
F=〈sinx,−3cosy,5xz〉 and C is the path given by
r(t)=(-2t^3,-3t^2,3t) for 0≤t≤1

Evaluate double integral Z 2 0 Z 1 y/2 cos(x^2 )dx dy
(integral from 0 to 2)(integral from y/2 to 1) for cos(x^2) dx
dy

Use Green’s theorem to evaluate the integral: ∫(-x^2y)dx
+(xy^2)dy where C is the boundary of the region enclosed by y=
sqrt(9 − x^2) and the x-axis, traversed in the counterclockwise
direction.

Evaluate the line integral by the two following methods.
xy dx + x2y3 dy
C is counterclockwise around the triangle with vertices
(0, 0), (1, 0), and (1, 2)
(a) directly
(b) using Green's Theorem

Evaluate the line integral by the two following methods.
(xy dx + x2 dy)
C is counterclockwise around the rectangle with
vertices (0, 0), (2, 0), (2, 1), (0, 1)
(a) directly
(b) using Green's Theorem

Evaluate the following integral. ∫ (x3 − 3x2)( 1 x −
3) dx
(A) − 3 4 x4 + 14 3 x3 − 3 2 x2 + C
(B) − 3 4 x4 + 4 x3 − 3x2 + C
(C) − 3 4 x4 + 10 3 x3 − 3x2 + C
(D) − 3 4 x4 + 10 3 x3 − 3 2 x2 + C
(E) ( 1 4 x4 − x3)( 1 x − 3) + ...

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