Application of Integrals: Class 12 Important Questions with Answers

Practice set: test yourself

Try every question before looking at the answers below. 64 or more out of 80 means you are ready for the board.

C. Very short answer (1 mark each)

1. Area under $y=3$ from $x=1$ to $x=5$.
2. Area of the region $x^2+y^2\le 16$, $x\ge 0$, $y\ge 0$.
3. Area under $y=\sqrt{x}$ from 0 to 1.
4. Area under $y=\sin 2x$ from 0 to $\frac{\pi }{2}$.
5. Area bounded by $y=x^3$ and the axis from 0 to 2.
6. Area under $y=\frac{1}{x}$ from 1 to $e$.
7. Area of $\frac{x^2}{9}+\frac{y^2}{16}=1$.
8. Area under $y=e^x$ from 0 to 2.
9. Area bounded by $y=|x|$, $x=-2$, $x=2$.
10. Area under $y=x^2+1$ from 0 to 1.

D. Short answer I (2 marks each)

1. Find the area bounded by $y=x^2$, the $y$-axis and $y=4$ in the first quadrant.
2. Find the area of the region bounded by $y=2x+1$, $x$-axis, $x=0$ and $x=2$.
3. Find the area under $y=\cos x$ from 0 to $\pi$.
4. Find the area bounded by $y^2=9x$ and $x=1$.
5. Find the area under $y=|x-2|$ from 0 to 4.
6. Find the area of the circle $x^2+y^2=25$ by integration.
7. Find the area under $y=3x^2$ between $x=-1$ and $x=1$.
8. Find the area bounded by $y=x-1$, the $x$-axis, $x=0$ and $x=2$.

E. Short answer II (3 marks each)

1. Find the area of the ellipse $9x^2+4y^2=36$.
2. Find the area bounded by $y=2\sqrt{x}$, the $x$-axis, $x=1$ and $x=4$.
3. Find the area of the region in the first quadrant enclosed by $y=x$, the $x$-axis and $x^2+y^2=8$ (to the right of $y=x$).
4. Sketch $y=|x-1|+1$ and find the area under it from $x=-1$ to $x=3$.
5. Find the area bounded by $x=4y^2$ (both halves) and the line $x=4$.
6. Find the area between $x^2+y^2=4$ and the lines $x=1$, $x=2$ above the $x$-axis.

F. Long answer (5 marks each)

1. Using integration, find the area of the triangle with vertices $(1,0)$, $(2,2)$ and $(3,1)$.
2. Using integration, find the area of the region $\{(x,y):x^2\le y\le 4\}$.

G. HOTS (higher-order thinking)

1. Why does ${\int }_0^{2\pi }\sin xdx=0$ while the area is 4?
2. Why is the area of an ellipse $\pi ab$ a natural extension of $\pi r^2$?
3. When is it easier to use horizontal strips?
4. Can an area be found by integration if the curve is not continuous?

20 MCQs

1. Area of $x^2+y^2=4$
   (a) $2\pi$ (b) $8\pi$ (c) $4\pi$ (d) $\pi$
2. Area of $\frac{x^2}{9}+\frac{y^2}{4}=1$
   (a) $6\pi$ (b) $13\pi$ (c) $36\pi$ (d) $5\pi$
3. Area under $y=x^2$ from 0 to 1
   (a) 1 (b) 3 (c) $\frac{1}{2}$ (d) $\frac{1}{3}$
4. Area under $y=\sin x$ from 0 to $\pi$
   (a) 0 (b) 2 (c) 1 (d) $\pi$
5. Area of the region $y=x$, $x$-axis, $x=2$
   (a) 4 (b) 2 (c) 1 (d) 8
6. Area bounded by $y=|x|$, $x=-1$, $x=1$
   (a) 0 (b) $\frac{1}{2}$ (c) 2 (d) 1
7. Area under $y=e^x$ from 0 to 1
   (a) $e-1$ (b) $e$ (c) $e+1$ (d) 1
8. Area bounded by $y^2=x$ and $x=1$
   (a) $\frac{2}{3}$ (b) 1 (c) $\frac{4}{3}$ (d) 2
9. Area under $y=\frac{1}{x}$ from 1 to $e^2$
   (a) 1 (b) $e^2$ (c) $e$ (d) 2
10. Area under $y=3$ from 0 to 2
    (a) 3 (b) 6 (c) 5 (d) 9
11. Area of quarter circle of radius 2
    (a) $2\pi$ (b) $4\pi$ (c) $\pi$ (d) $\frac{\pi }{2}$
12. Area under $y=x^3$ from $-1$ to 1
    (a) $\frac{1}{2}$ (b) 0 (c) 1 (d) 2
13. Area under $y=\cos x$ from 0 to $\pi$
    (a) 0 (b) 2 (c) 1 (d) $\pi$
14. Area between $x=y^2$ and $y$-axis, $0\le y\le 3$
    (a) 3 (b) 27 (c) 9 (d) 6
15. Area under $y=2x$ from 1 to 3
    (a) 4 (b) 6 (c) 9 (d) 8
16. Area under $y=\sqrt{4-x^2}$ from 0 to 2
    (a) $\pi$ (b) $2\pi$ (c) $4\pi$ (d) $\frac{\pi }{2}$
17. Area of $4x^2+y^2=4$
    (a) $2\pi$ (b) $4\pi$ (c) $\pi$ (d) $8\pi$
18. Area under $y=|x-1|$ from 0 to 2
    (a) 2 (b) 0 (c) $\frac{1}{2}$ (d) 1
19. Area under $y=\sin x$ from 0 to $2\pi$
    (a) 0 (b) 4 (c) 2 (d) $2\pi$
20. Area of $y^2=4x$ and $x=1$
    (a) $\frac{4}{3}$ (b) $\frac{2}{3}$ (c) $\frac{8}{3}$ (d) 4

1 (c) · 2 (a) · 3 (d) · 4 (b) · 5 (b) · 6 (d) · 7 (a) · 8 (c) · 9 (d) · 10 (b) · 11 (c) · 12 (a) · 13 (b) · 14 (c) · 15 (d) · 16 (a) · 17 (a) · 18 (d) · 19 (b) · 20 (c)

Assertion–Reason (6)

(a) Both true, R explains A. (b) Both true, R does not explain A. (c) A true, R false. (d) A false, R true.

1. A: Area bounded by $y=x^3$, $x=-1$, $x=1$ is $\frac{1}{2}$. R: area takes $|y|$, adding the parts on both sides.
2. A: Area of $x^2+y^2=a^2$ is $\pi a^2$. R: ${\int }_0^a\sqrt{a^2-x^2}dx=\frac{\pi a^2}{4}$.
3. A: ${\int }_0^{2\pi }\cos xdx$ is the area under $\cos x$ on $[0,2\pi ]$. R: ${\int }_0^{2\pi }\cos xdx=0$.
4. A: Area of $\frac{x^2}{4}+\frac{y^2}{9}=1$ is $6\pi$. R: area of an ellipse is $\pi ab$.
5. A: The area under $y=x^2$ from 0 to 2 equals the area between $x=\sqrt{y}$ and the $y$-axis from 0 to 4. R: both are $\frac{8}{3}$.
6. A: Area under $y=\frac{1}{x}$ from 1 to 2 is $\log 2$. R: $\int \frac{dx}{x}=\log |x|$.

Key: 1 (a) · 2 (a) · 3 (d) · 4 (a) · 5 (d) · 6 (a)

Answers to the practice set