The area bounded by the curves $y = | x ^ { 2 } - 1 |$ and $y = 1$ is
(1) $\frac { 2 } { 3 } ( \sqrt { 2 } + 1 )$
(2) $\frac { 4 } { 3 } ( \sqrt { 2 } - 1 )$
(3) $2 ( \sqrt { 2 } - 1 )$
(4) $\frac { 8 } { 3 } ( \sqrt { 2 } - 1 )$

If the tangent at a point P on the parabola $\mathrm { y } ^ { 2 } = 3 \mathrm { x }$ is parallel to the line $x + 2 y = 1$ and the tangents at the points $Q$ and $R$ on the ellipse $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 1 } = 1$ are perpendicular to the line $x - y = 2$, then the area of the triangle $P Q R$ is: (1) $\frac { 9 } { \sqrt { 5 } }$ (2) $5 \sqrt { 3 }$ (3) $\frac { 3 } { 2 } \sqrt { 5 }$ (4) $3 \sqrt { 5 }$

The area of the region enclosed by the curve $y = x ^ { 3 }$ and its tangent at the point $( - 1 , - 1 )$ is
(1) $\frac { 19 } { 4 }$
(2) $\frac { 23 } { 4 }$
(3) $\frac { 31 } { 4 }$
(4) $\frac { 27 } { 4 }$

If the area enclosed by the parabolas $P _ { 1 } : 2 y = 5 x ^ { 2 }$ and $P _ { 2 } : x ^ { 2 } - y + 6 = 0$ is equal to the area enclosed by $P _ { 1 }$ and $y = \alpha x , \alpha > 0$, then $\alpha ^ { 3 }$ is equal to $\_\_\_\_$.

Let $A$ be the area of the region $\left\{(x, y): y \geq x^{2},\, y \geq (1-x)^{2},\, y \leq 2x(1-x)\right\}$. Then $540A$ is equal to

Q76. One of the points of intersection of the curves $y = 1 + 3 x - 2 x ^ { 2 }$ and $y = \frac { 1 } { x }$ is $\left( \frac { 1 } { 2 } , 2 \right)$. Let the area of the region enclosed by these curves be $\frac { 1 } { 24 } ( l \sqrt { 5 } + \mathrm { m } ) - \mathrm { n } \log _ { \mathrm { e } } ( 1 + \sqrt { 5 } )$, where $l , \mathrm {~m} , \mathrm { n } \in \mathbf { N }$. Then $l + \mathrm { m } + \mathrm { n }$ is equal to
(1) 29
(2) 31
(3) 30
(4) 32

Q89. The area of the region enclosed by the parabolas $y = x ^ { 2 } - 5 x$ and $y = 7 x - x ^ { 2 }$ is

3. (a) Find the coordinates of the points at which the two curves $y = 6 x ^ { 2 }$ and $y = x ^ { 4 } - 16$ intersect.
(b) Give a rough sketch of the two curves (in the same diagram) for the range $- 3 \leq x \leq 3$.
(c) Find the area of the region enclosed by the two curves.

Calculate the area of the region bounded by the graphs of the functions
$$f ( x ) = 2 + x - x ^ { 2 } , \quad g ( x ) = 2 x ^ { 2 } - 4 x$$

Given the functions
$$f ( x ) = 2 + 2 x - 2 x ^ { 2 } \text { and } g ( x ) = 2 - 6 x + 4 x ^ { 2 } + 2 x ^ { 3 } ,$$
find:\ a) (1 point) Study the differentiability of $h ( x ) = | f ( x ) |$.\ b) (1.5 points) Find the area of the region bounded by the curves\ $y = f ( x ) , y = g ( x ) , x = 0$ and $x = 2$.

On the coordinate plane, let $\Gamma$ denote the graph of the polynomial function $y = x ^ { 3 } - 4 x ^ { 2 } + 5 x$ , and let $L$ denote the line $y = m x$ , where $m$ is a real number. Based on the above, answer the following questions.
(1) When $m = 2$ , find the $x$-coordinates of the three distinct intersection points of $\Gamma$ and $L$ in the range $x \geq 0$ . (2 points)
(2) Based on (1), find the area of the bounded region enclosed by $\Gamma$ and $L$ . (4 points)
(3) In the range $x \geq 0$ , if $\Gamma$ and $L$ have three distinct intersection points, then the maximum range of $m$ satisfying this condition is $a < m < b$ . Find the values of $a$ and $b$ . (6 points)

As shown in the figure, the Wang family owns a triangular piece of land $\triangle A B C$ , where $\overline { B C } = 16$ meters. The government plans to requisition the trapezoid $D B C E$ portion to develop a road with straight lines $D E , B C$ as edges, with road width $h$ meters, leaving the Wang family with only $\frac { 9 } { 16 }$ of the original land area. After negotiation, the plan is changed to develop a road with parallel lines $B E , F C$ as edges, with the same road width, where $\angle E B C = 30 ^ { \circ }$ . Only the $\triangle B C E$ region needs to be requisitioned. According to this agreement, the Wang family's remaining land $\triangle A B E$ has (15-1)(15-2)(15-3) square meters.

Let $a , b$ be real numbers, and let $O$ be the origin of the coordinate plane. It is known that the graph of the quadratic function $f ( x ) = a x ^ { 2 }$ and the circle $\Omega : x ^ { 2 } + y ^ { 2 } - 3 y + b = 0$ both pass through point $P \left( 1 , \frac { 1 } { 2 } \right)$, and let point $C$ be the center of $\Omega$.
Find the area of the region bounded by the graph of $y = f ( x )$ above and the lower semicircular arc of $\Omega$.

$p$ is a positive constant. Find the area enclosed between the curves $y = p\sqrt{x}$ and $x = p\sqrt{y}$

The area enclosed between the line $y = mx$ and the curve $y = x^3$ is 6.
What is the value of $m$?
A $2$
B $4$
C $\sqrt{3}$
D $\sqrt{6}$
E $2\sqrt{3}$
F $2\sqrt{6}$

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) $\frac{2}{3}$

The function $f$ is one-to-one, and the shaded region between the lines $y = x$ and $x = 1$ and the curve $y = f ( x )$ in the first quadrant is given below.
Which of the following is the expression of the area of the shaded region in terms of $\mathbf { f } ^ { - \mathbf { 1 } } ( \mathbf { x } )$?
A) $\int _ { 0 } ^ { 2 } f ^ { - 1 } ( x ) d x$
B) $\int _ { 0 } ^ { 2 } \left( 2 - f ^ { - 1 } ( x ) \right) d x$
C) $\int _ { 0 } ^ { 1 } \left( x - f ^ { - 1 } ( x ) \right) d x$
D) $\int _ { 0 } ^ { 1 } \left( 2 - f ^ { - 1 } ( x ) \right) d x + \int _ { 1 } ^ { 2 } f ^ { - 1 } ( x ) d x$
E) $\int _ { 0 } ^ { 1 } \left( x - f ^ { - 1 } ( x ) \right) d x + \int _ { 1 } ^ { 2 } \left( 1 - f ^ { - 1 } ( x ) \right) d x$

In the rectangular coordinate plane; the region between the curve $y = 3 \sqrt { x }$, the line $x = 1$, and the line $y = 0$ is divided into two regions of equal area by the line $y = m x$.
Accordingly, what is m?
A) $\frac { 3 } { 2 }$
B) $\frac { 4 } { 3 }$
C) $\frac { 5 } { 4 }$
D) 1
E) 2

Let k be a positive real number. The area of the bounded region between the line $\mathrm { y } = \mathrm { kx }$ and the parabola $y = x ^ { 2 }$ is $\frac { 9 } { 16 }$ square units.
Accordingly, what is the value of $\mathbf { k }$?
A) $\frac { 3 } { 2 }$
B) $\frac { 4 } { 3 }$
C) $\frac { 7 } { 4 }$
D) $\frac { 7 } { 6 }$
E) $\frac { 8 } { 5 }$

In the Cartesian coordinate plane, the graphs of functions $f$, $g$ and $h$ are shown below.
The areas of the shaded regions A1, A2 and A3 shown in the figure are 1, 3 and 9 square units, respectively.
Accordingly, $$\int _ { a } ^ { c } ( h ( x ) - g ( x ) ) d x + \int _ { b } ^ { d } ( f ( x ) - h ( x ) ) d x$$
what is the value of the integral?
A) 5 B) 8 C) 12 D) 13 E) 17

In the rectangular coordinate plane,
$$\begin{aligned} & f ( x ) = x ^ { 2 } - 2 x \\ & g ( x ) = - x ^ { 2 } + 4 x \end{aligned}$$
The shaded region between the graphs of these functions and the x-axis is given below.
Accordingly, what is the area of the shaded region in square units?
A) $\frac { 17 } { 3 }$
B) $\frac { 19 } { 3 }$
C) $\frac { 20 } { 3 }$
D) $\frac { 22 } { 3 }$
E) $\frac { 23 } { 3 }$