Areas of regions bounded by simple curves — practice questions

Question 1 · difficulty L2 · understanding

The area of the region bounded by the curves $x+3 y^2=0$ and $x+4 y^2=1$ is equal to :

• A. $\frac{1}{3}$
• B. $\frac{2}{3}$
• C. $\frac{4}{3}$
• D. $\frac{5}{3}$

Question 2 · difficulty L2 · understanding

The area of the region : $$R = \{ (x,y):5{x^2} \le y \le 2{x^2} + 9\}$$ is :

• A. $$6\sqrt 3$$ square units
• B. $$12\sqrt 3$$ square units
• C. $$11\sqrt 3$$ square units
• D. $$9\sqrt 3$$ square units

Question 3 · difficulty L2 · understanding

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

Question 4 · difficulty L2 · understanding

A farmer F 1 has a land in the shape of a triangle with vertices at P(0, 0), Q(1, 1) and R(2, 0). From this land, a neighbouring farmer F 2 takes away the region which lies between the sides PQ and a curve of the form y = x n (n > 1). If the area of the region taken away by the farmer F 2 is exactly 30% of the area of $$\Delta$$PQR, then the value of n is .................

Question 5 · difficulty L3 · understanding

The area of the region $\left\{(x, y): 0 \leq y \leq 6-x, y^2 \geq 4 x-3, x \geq 0\right\}$ is :

• A. 8
• B. 9
• C. 12
• D. 15

Question 6 · difficulty L3 · understanding

The area of the region $\mathrm{R}=\left\{(x, y): x y \leq 27,1 \leq y \leq x^2\right\}$ is equal to :

• A. $$78 \log _e 3-\frac{52}{3}$$
• B. $$54 \log _e 3-\frac{52}{3}$$
• C. $$54 \log _e 3-\frac{26}{3}$$
• D. $$54 \log _e 3+\frac{26}{3}$$

Question 7 · difficulty L3 · understanding

Let $P_1 : y = 4x^2$ and $P_2 : y = x^2 + 27$ be two parabolas. If the area of the bounded region enclosed between $P_1$ and $P_2$ is six times the area of the bounded region enclosed between the line $y = \alpha x$, $\alpha > 0$ and $P_1$, then $\alpha$ is equal to :

• A. 12
• B. 15
• C. 8
• D. 6

Question 8 · difficulty L3 · understanding

The area of the region $\mathrm{R}=\left\{(x, y): x y \leq 8,1 \leq y \leq x^2, x \geq 0\right\}$ is

• A. $\frac{2}{3}\left(20 \log _e(2)+9\right)$
• B. $\frac{1}{3}\left(40 \log _e(2)+27\right)$
• C. $\frac{1}{3}\left(49 \log _e(2)-15\right)$
• D. $\frac{2}{3}\left(24 \log _e(2)-7\right)$

Question 9 · difficulty L3 · understanding

Let $\mathrm{A}_1$ be the bounded area enclosed by the curves $y=x^2+2, x+y=8$ and $y$-axis that lies in the first quadrant. Let $\mathrm{A}_2$ be the bounded area enclosed by the curves $y=x^2+2, y^2=x, x=2$, and $y$-axis that lies in the first quadrant. Then $\mathrm{A}_1-\mathrm{A}_2$ is equal to

• A. $\frac{2}{3}(2 \sqrt{2}+1)$
• B. $\frac{2}{3}(3 \sqrt{2}+1)$
• C. $\frac{2}{3}(\sqrt{2}+1)$
• D. $\frac{2}{3}(4 \sqrt{2}+1)$

Question 10 · difficulty L3 · understanding

The area of the region enclosed between the circles $x^2+y^2=4$ and $x^2+(y-2)^2=4$ is:

• A. $\frac{2}{3}(4 \pi-3 \sqrt{3})$
• B. $\frac{4}{3}(2 \pi-\sqrt{3})$
• C. $\frac{4}{3}(2 \pi-3 \sqrt{3})$
• D. $\frac{2}{3}(2 \pi-3 \sqrt{3})$