How are triple integrals defined in cylindrical and spherical coordinates? Why might one prefer working in one of these coordinate systems to working in rectangular coordinates?

To determine:
a) Whether the statement, " The point with Cartesian coordinates ${\displaystyle \left[\begin{array}{cc}-2& 2\end{array}\right]}$ has polar coordinates ${\displaystyle \left[bf(2\sqrt{2},\frac{3\pi }{4})(2\sqrt{2},\frac{11\pi }{4})(2\sqrt{2},-\frac{5\pi }{4})\text{and}(-2\sqrt{2},-\frac{\pi }{4})\right]}$ " is true or false.
b) Whether the statement, " the graphs of ${\displaystyle [r\cos \theta =4\text{and}r\sin \theta =-2]}$ intersect exactly once " is true or false.
c) Whether the statement, " the graphs of ${\displaystyle [r=4\text{and}\theta =\frac{\pi }{4}]}$ intersect exactly once ", is true or false.
d) Whether the statement, " the point ${\displaystyle \left[\begin{array}{cc}3& \frac{\pi }{2}\end{array}\right]liesonthegraphof[r=3\cos 2\theta ]}$ " is true or false.
e) Whether the statement, " the graphs of ${\displaystyle [r=2\sec \theta \text{and}r=3\csc \theta ]}$ are lines " is true or false.

Given the full and correct answer the two bases of $B1=\{\left(\begin{array}{c}2\\ 1\end{array}\right),\left(\begin{array}{c}3\\ 2\end{array}\right)\}$
$B_2=\{\left(\begin{array}{c}3\\ 1\end{array}\right),\left(\begin{array}{c}7\\ 2\end{array}\right)\}$
find the change of basis matrix from ${\displaystyle B_1\to B_2}$ and next use this matrix to covert the coordinate vector
${\overrightarrow{v}}_{B_1}=\left(\begin{array}{c}2\\ -1\end{array}\right)$ of v to its coodirnate vector
${\overrightarrow{v}}_{B_2}$

(10%) In $R^2$, there are two sets of coordinate systems, represented by two distinct bases: $(x_1,y_1)$ and $(x_2,y_2)$. If the equations of the same ellipse represented by the two distinct bases are described as follows, respectively: $2(x_1{)}^2-4(x_1)(y_1)+5(y_1{)}^2-36=0$ and $(x_2{)}^2+6(y_2{)}^2-36=0.$ Find the transformation matrix between these two coordinate systems: $(x_1,y_1)$ and $(x_2,y_2)$.