Sketch the solid obtained by rotating the region underneath the graph of the function over the given interval about the y-axis, and find its volume.

$$f ( x ) = x ^ { 3 },$$

[0, 1]

Write the system of equations represented by the matrix equation below. Then solve matrix equation.

$$\left[ \begin{array} { r r } { - 3 } & { - 2 } \\ { 5 } & { 4 } \end{array} \right] \left[ \begin{array} { l } { x } \\ { y } \end{array} \right] = \left[ \begin{array} { r } { 1 } \\ { - 3 } \end{array} \right]$$

Evaluate the limits or explain why the limit fails to exist.

$$\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { ( x + y ) ^ { 2 } } { x ^ { 2 } + y ^ { 2 } }$$

Simplify the expression.

$$\left( a ^ { 2 } \right) ^ { - 3 } \left( a ^ { 3 } b \right) ^ { 2 } \left( b ^ { 3 } \right) ^ { 4 }$$

(a) verify the given factor(s) of the function f, (b) find the remaining factors of f, (c) use your results to write the complete factorization of f, and (d) list all real zeros of f. Confirm your results by using a graphing utility to graph the function. Function $f(x)=2 x^{3}-x^{2}-10 x+5$ Factor(s) $(2x - 1)$

Hugo has a small print of one of the paintings in the table. It is similar in size to the original. The print measures 11 in. $\times$ 10 in. Of which painting is this a print? Explain.

$$\begin{matrix} & & \text{Original}\\\text{Painting} & \text{Artist} & \text{Size (in.)}\\\text{Mona Lisa} & \text{Leonardo Da Vinci} & \mathrm{30 \times 21}\\\text{The Dance Class} & \text{Edgar Degas} & \mathrm{33 \times 30}\\ \text{The Blue Vase} & \text{Paul Cezanne} & \mathrm{28 \times 18}\\\end{matrix}$$

Create a three-column chart with these headings: Term, Original Meaning, and Meaning in English. Then, using a dictionary that contains etymologies, fill in the chart for each of the following terms. coup de grâce

Rewrite the formula for the area of a rhombus for the special case of a square with side length s. Show that this is the same as the formula for the area of a square, A = s².

$$\text { Find all monic irreducible polynomials of degree } 2 \text { over } Z _ { 3 }$$

$$\text{ Consider the following joint probability density function of the random variables } X \text{ and } Y :$$

$$f(x,y)=\frac{3x-y}{9}, \text{ for } 1<x<3, \text{ } 1<y<2,$$

$$f(x,y)=0, \text{ elsewhere.}$$

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$$(a) \text{ Find the marginal density functions of } X \text{ and } Y .$$

$$(b) \text{ Are } X \text{ and } Y \text{ independent?}$$

$$(c) \text{ Find } P(X >2).$$

Evaluate the expression. 8m, if m = -5

An equation of an ellipse is given. Sketch a graph of the ellipse. $\frac{x^{2}}{4}+\frac{y^{2}}{36}=1$

A hockey puck moving at 0.45 m/s collides with another puck that was at rest. The pucks have equal mass. The first puck is deflected

$$37 ^ { \circ }$$

to the right and moves off at 0.36 m/s. Find the speed and direction of the second puck after the collision.

Determine the limit of the sequence or show that the sequence diverges.

$$a _ { n } = \frac { 10 ^ { n } } { n ! }$$