A área da região delimitada pela parábola $y = x^2$ e pela reta $y = 4$ é
(A) $\dfrac{16}{3}$ (B) $\dfrac{32}{3}$ (C) $8$ (D) $\dfrac{64}{3}$ (E) $16$

The area enclosed by the curve $y = - 2 x ^ { 2 } + 3 x$ and the line $y = x$ is $\frac { q } { p }$. Find the value of $p + q$. (Here, $p$ and $q$ are coprime natural numbers.) [4 points]

Find the area enclosed by the curve $y = x ^ { 2 } - 7 x + 10$ and the line $y = - x + 10$. [4 points]

What is the area of the region enclosed by the two curves $y = x ^ { 2 } + 3$, $y = - \frac { 1 } { 5 } x ^ { 2 } + 3$ and the line $x = 2$? [3 points]
(1) $\frac { 18 } { 5 }$
(2) $\frac { 7 } { 2 }$
(3) $\frac { 17 } { 5 }$
(4) $\frac { 33 } { 10 }$
(5) $\frac { 16 } { 5 }$

Consider the polynomial
$$f ( x ) = 1 + 2 x + 3 x ^ { 2 } + 4 x ^ { 3 }$$
Let s be the sum of all distinct real roots of $\mathrm { f } ( \mathrm { x } )$ and let $\mathrm { t } = | \mathrm { s } |$.
The area bounded by the curve $y = f ( x )$ and the lines $x = 0 , y = 0$ and $x = t$, lies in the interval
A) $\left( \frac { 3 } { 4 } , 3 \right)$
B) $\left( \frac { 21 } { 64 } , \frac { 11 } { 16 } \right)$
C) $( 9,10 )$
D) $\left( 0 , \frac { 21 } { 64 } \right)$

Let $g ( x ) = \left( x - \frac { 1 } { 2 } \right) ^ { 2 } , x \in R$. Then, the area (in sq. units) of the region bounded by the curves, $y = f ( x )$ and $y = g ( x )$ between the lines $2 x = 1$ and $2 x = \sqrt { 3 }$, is:
(1) $\frac { 1 } { 3 } + \frac { \sqrt { 3 } } { 4 }$
(2) $\frac { \sqrt { 3 } } { 4 } - \frac { 1 } { 3 }$
(3) $\frac { 1 } { 2 } - \frac { \sqrt { 3 } } { 4 }$
(4) $\frac { 1 } { 2 } + \frac { \sqrt { 3 } } { 4 }$

Let the area of the region enclosed by the curves $y = 3 x , 2 y = 27 - 3 x$ and $y = 3 x - x \sqrt { x }$ be $A$. Then $10 A$ is equal to
(1) 172
(2) 162
(3) 154
(4) 184

The area of the region enclosed by the curves $y = x ^ { 2 } - 4 x + 4$ and $y ^ { 2 } = 16 - 8 x$ is :
(1) $\frac { 8 } { 3 }$
(2) $\frac { 4 } { 3 }$
(3) 8
(4) 5

The area of the region bounded by the curves $x \left( 1 + y ^ { 2 } \right) = 1$ and $y ^ { 2 } = 2 x$ is:
(1) $2 \left( \frac { \pi } { 2 } - \frac { 1 } { 3 } \right)$
(2) $\frac { \pi } { 2 } - \frac { 1 } { 3 }$
(3) $\frac { \pi } { 4 } - \frac { 1 } { 3 }$
(4) $\frac { 1 } { 2 } \left( \frac { \pi } { 2 } - \frac { 1 } { 3 } \right)$

What is the area in square units of the (finite) region bounded by the curve $y = x^{3}$ and the line $y = x$?
A) $\frac{1}{2}$
B) $\frac{3}{2}$
C) $1$
D) $\frac{1}{3}$
E) $\frac{2}{3}$