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3 years ago

8

Use the given transformation to evaluate the integral. (x ? 10y) dA, R where R is the triangular region with vertices (0, 0), (9

, 1), and (1, 9). x = 9u + v, y = u + 9v

1 answer:

lions [1.4K]3 years ago

4 0

The given change of coordinates has Jacobian

$\mathbf J=\begin{bmatrix}\dfrac{\partial x}{\partial u}&\dfrac{\partial x}{\partial v}\\\\\dfrac{\partial y}{\partial u}&\dfrac{\partial y}{\partial v}\end{bmatrix}=\begin{bmatrix}9&1\\1&9\end{bmatrix}$

so the area element is

$\mathrm dA=\mathrm dx\,\mathrm dy=|\det\mathbf J|\,\mathrm du\,\mathrm dv=80\,\mathrm du\,\mathrm dv$

The new region is a right triangle with vertices (0, 0), (1, 0), and (0, 1) in the $u,v$ plane.

Then the integral becomes

$\displaystyle\iint_R(x-10y)\,\mathrm dA=-80\int_{u=0}^{u=1}\int_{v=0}^{v=1-u}(u+89v)\,\mathrm dv\,\mathrm du=-1200$

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See explanation

Step-by-step explanation:

Q1-5.

1. Plane parallel to WXT is ZYU.

2. Segments parallel to $\overline {VU}$ are $\overline {ZY}, \overline {WX}$ and $\overline {ST}$

3. Segments parallel to $\overline {SW}$ are $\overline {VZ}, \overline {YU}$ and $\overline {XT}$

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5. Segments skew to $\overline {}$ $\overline {VZ}$ are $\overline {WX}$ and $\overline {XT}$ (not lie in the same plane and not parallel)

Q6.

a. $\angle 4$ and $\angle 10$ are the same-side interior angles, transversal k

b. $\angle 8$ and $\angle 11$ are alternate exterior angles, transversal m

c. $\angle 1$ and $\angle 4$ do not form any pair of angles

d. $\angle 2$ and $\angle 12$ are the same-side exterior angles, transversal k

e. $\angle 5$ and $\angle 7$ are corresponding angles, transversal j

f. $\angle 2$ and $\angle 13$ are alternate interior angles, transversal l

Q7.

$m\angle 1=m\angle 7=131^{\circ}$ (as vertical angle with angle 7)

$m\angle 2=180^{\circ}-131^{\circ}=49^{\circ}$ (as supplementary angle with angle 1)

$m\angle 8=49^{\circ}$ (as vertical angle with angle 2)

$m\angle 3=m\angle 1=131^{\circ}$ (as corresponding angles when parallel lines p and q are cut by transversal r)

$m\angle 4=m\angle 2=49^{\circ}$ (as corresponding angles when parallel lines p and q are cut by transversal r)

$m\angle 5=m\angle 7=131^{\circ}$ (as corresponding angles when parallel lines p and q are cut by transversal r)

$m\angle 6=m\angle 8=49^{\circ}$ (as corresponding angles when parallel lines p and q are cut by transversal r)

$m\angle 10=m\angle 16=88^{\circ}$ (as vertical angle with angle 16)

$m\angle 9=180^{\circ}-88^{\circ}=92^{\circ}$ (as supplementary angle with angle 16)

$m\angle 15=92^{\circ}$ (as vertical angle with angle 9)

$m\angle 14=m\angle 16=88^{\circ}$ (as corresponding angles when parallel lines p and q are cut by transversal s)

$m\angle 13=m\angle 15=92^{\circ}$ (as corresponding angles when parallel lines p and q are cut by transversal s)

$m\angle 12=m\angle 10=88^{\circ}$ (as corresponding angles when parallel lines p and q are cut by transversal s)

$m\angle 11=m\angle 9=92^{\circ}$ (as corresponding angles when parallel lines p and q are cut by transversal s)

Q8.

$m\angle 7=m\angle 9=105^{\circ}$ (as vertical angles)

$m\angle 8=180^{\circ}-105^{\circ}=75^{\circ}$ (as supplementary angle with angle 9)

$m\angle 10=m\angle 8=75^{\circ}$ (as vertical angles)

$m\angle 6=m\angle 8=75^{\circ}$ (as alternate interior angles when parallel lines a and b are cut by transversal c)

$m\angle 1=180^{\circ}-75^{\circ}-63^{\circ}=42^{\circ}$ (by angle addition postulate)

$m\angle 3=180^{\circ}-42^{\circ}-63^{\circ}=75^{\circ}$ (by angle addition postulate)

$m\angle 4=m\angle 1=42^{\circ}$ (as vertical angles)

$m\angle 5=m\angle 2=63^{\circ}$ (as vertical angles)

$m\angle 11=m\angle 4=42^{\circ}$ (as alternate interior angles when parallel lines a and b are cut by transversal d)

$m\angle 12=180^{\circ}-42^{\circ}=138^{\circ}$ (as supplementary angles)

$m\angle 13=m\angle 11=42^{\circ}$ (as vertical angles)

$m\angle 14=m\angle 12=138^{\circ}$ (as vertical angles)

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