Question

Consider the double integral R ???(? − ?) ???(? + ?) ?? ? where ? is the triangle in the ??-plane with vertices at (0,0), (π, −π), , and (π. π).

a) Let ? = ? − ? and ? = ? + ?. Sketch the region of integration

b) Find ?(?, ?).

c) Use the change of variables to calculate the integration. (Hint: Trig functions are 2?-periodic and you will need half-angle identities at some point)

Answer #1

Evaluate the integral ∬ ????, where ? is the square with
vertices (0,0),(1,1), (2,0), and (1,−1), by carrying out the
following steps:
a. sketch the original region of integration R in the xy-plane
and the new region S in the uv-plane using this variable change: ?
= ? + ?,? = ? − ?,
b. find the limits of integration for the new integral with
respect to u and v,
c. compute the Jacobian,
d. change variables and evaluate the...

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...

Evaluate the given integral by making an appropriate change of
variables, where R is the trapezoidal region with vertices (3, 0),
(4, 0), (0, 4), and (0, 3).
L = double integral(7cos(7(x-y)/(x+y))dA

Use the given transformation to evaluate the double integral of
(x-6y) dA, where R is the triangular region with vertices
(0, 0), (5, 1), and (1, 5).
x = 5u +
v, y = u +
5v

1. Let R be the rectangle in the xy-plane bounded by the lines x
= 1, x = 4, y = −1, and y = 2. Evaluate Z Z R sin(πx + πy) dA.
2. Let T be the triangle with vertices (0, 0), (0, 2), and (1,
0). Evaluate the integral Z Z T xy^2 dA
ZZ means double integral. All x's are variables. Thank you!.

2. Consider the line integral I C F · d r, where the vector
field F = x(cos(x 2 ) + y)i + 2y 3 (e y sin3 y + x 3/2 )j and C is
the closed curve in the first quadrant consisting of the curve y =
1 − x 3 and the coordinate axes x = 0 and y = 0, taken
anticlockwise.
(a) Use Green’s theorem to express the line integral in terms of
a double...

We consider the plane region R delimited by the curves y = cos (x) and y = (x − π) ^ 2 −2.
(a) Determine the volume of the solid generated by the rotation of R revolves around the
right y = −3.
(b) Determine the volume of the solid generated by the rotation of R revolves around the
right x = 0.
For (a) and (b), observe the following procedure:
- Draw a sketch (2D) of the R region...

Consider the nonlinear second-order differential
equation 4x"+4x'+2(k^2)(x^2)− 1/2 =0, where k > 0 is a constant.
Answer to the following questions.
(a) Show that there is no periodic solution in a simply connected
region R={(x,y) ∈ R2 | x <0}. (Hint: Use the corollary to
Theorem 11.5.1>>
If symply connected region R either contains no critical points of
plane autonomous system or contains a single saddle point, then
there are no periodic solutions. )
(b) Derive a plane autonomous system...

Read the attached articles about the proposed merger of Xerox
and Fujifilm. Utilizing your knowledge of external and internal
analysis, business and corporate strategy, and corporate
governance, please discuss the following questions:
1. What is the corporate strategy behind the merger of Xerox and
Fujifilm?
2. Why did Xerox agree to the merger? Is this a good deal for
Xerox? Discuss the benefits and challenges they face with the
merger.
3. Why did Fujifilm agree to the merger? Discuss the...

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