Question

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

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 the integral by reversing the order of integration.
2
0
2
6ex/y dy dx
x

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 ∮C(x^3+xy)dx+(cos(y)+x2)dy∮C(x^3+xy)dx+(cos(y)+x^2)dy
where C is the positively oriented boundary of the region bounded
by C:0≤x^2+y^2≤16, x≥0,y≥0C:0≤x^2+y^2≤16,x≥0,y≥0

Evaluate the line integral of " (y^2)dx +
(x^2)dy " over the closed curve C which is the triangle
bounded by x = 0, x+y = 1, y = 0.

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.

Evaluate the line integral:
(x^2 + y^2) dx + (5xy) dy on the edge of the circle: x^2 + y^2 = 4.
USING GREEN'S THEOREM.

Evaluate C (y + 6 sin(x)) dx + (z2 + 2 cos(y)) dy + x3 dz where
C is the curve r(t) = sin(t), cos(t), sin(2t) , 0 ≤ t ≤ 2π. (Hint:
Observe that C lies on the surface z = 2xy.) C F · dr =

2. Evaluate the double integral Z Z R e ^(x^ 2+y ^2) dA where R
is the semicircular region bounded by x ≥ 0 and x^2 + y^2 ≤ 4.
3. Find the volume of the region that is bounded above by the
sphere x^2 + y^2 + z^2 = 2 and below by the paraboloid z = x^2 +
y^2 .
4. Evaluate the integral Z Z R (12x^ 2 )(y^3) dA, where R is the
triangle with vertices...

find dx/dx and dz/dy
z^3 y^4 - x^2 cos(2y-4z)=4z

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