Question 1

Evaluate the integral by making an appropriate change of variables.Round your answer to two decimal places.   R is the parallelogram bounded by the lines

Accepted Answer

11ea8a3b_b740_9d1c_9fff_091ca6e34ea2_TB6897_11

Question 2

Use the given transformation to evaluate the integral.   ,where R is the square with vertices (0,0),(4,6),(6,   ),(10,2)and  &#10;A) 208&#10;B) 42&#10;C) 312&#10;D) 52&#10;E) 343

Accepted Answer

The given transformation is $T(x,y) = (2x-3y,x+y)$. We can apply this transformation to each of the vertices of the square to get the new vertices of the image square: 
\begin{align*}
T(0,0) &= (-3,0) \
T(4,6) &= (5,10) \
T(6,2) &= (3,8) \
T(10,4) &= (17,14)
\end{align*}
Thus, the image square has vertices $(-3,0)$, $(5,10)$, $(3,8)$, and $(17,14)$. We can now set up the integral in terms of this new square: 
$$\iint_R f(x,y)\, dA = \iint_{R'} f(T^{-1}(x,y))\left|\frac{\partial(x,y)}{\partial(u,v)}ight|\, dA$$
where $R'$ is the image square and $T^{-1}(x,y)$ is the inverse transformation given by $T^{-1}(u,v) = \left(\frac{u+v}{2},\frac{-u+3v}{2}ight)$. The Jacobian is $\left|\frac{\partial(x,y)}{\partial(u,v)}ight| = 2$.

We are given that $f(x,y) = x+y$, so we have 
\begin{align*}
\iint_R (x+y)\, dA &= \iint_{R'} (u+v)\cdot 2\, dA \
&= 2 \iint_{R'} (u+v)\, dA \
&= 2\left(\int_3^5\int_0^8 (u+v)\, dvdu + \int_{-3}^3 \int_0^{10} (u+v)\,dvduight) \
&= 2\left(\int_3^5 \left[uv + \frac{1}{2}v^2ight]_{v=0}^{v=8} du + \int_{-3}^3 \left[uv + \frac{1}{2}v^2ight]_{v=0}^{v=10}duight) \
&= 2\left(\int_3^5 (8u+32)\, du + \int_{-3}^3 (10u+50)\,duight) \
&= 2\left(\left[4u^2 + 32uight]_{u=3}^{u=5} + \left[5u^2 + 50uight]_{u=-3}^{u=3}ight) \
&= 2\left((116+64) + (45-15)ight) \
&= 2(210) \
&= \boxed{420} \
\end{align*}

Question 3

Find the Jacobian of the transformation.  &#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E)

Accepted Answer

11ea8a3b_b740_00ca_9fff_f7f64c02bef5_TB6897_11

Question 4

Use the transformation   to evaluate the integral   ,where R is the region bounded by the ellipse  &#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E)

Accepted Answer

The answer of Use the transformation   to evaluate...

Question 5

Use the given transformation to evaluate the integral.   ,where R is the region in the first quadrant bounded by the lines   and the hyperbolas  &#10;A) 4.447&#10;B) 3.296&#10;C) 5.088&#10;D) 9.447&#10;E) 8.841

Accepted Answer

The answer of Use the given transformation to evaluate the...

Question 6

Find the Jacobian of the transformation.

Accepted Answer

The answer of Find the Jacobian of the transformation. ...

Deck 12: Extension H: Multiple Integrals