Vidaara.orgClass 12 · Mathematics
CodeVID-M12-08-CT
Application of Integrals — Full Chapter Test
Name: ____________________
Roll No.: __________
Date: ____________
General Instructions
- This is a full-length test covering the whole chapter — every topic is included.
- All questions are compulsory.
- Section A carries 1 mark each, Section B 2 marks, Section C 3 marks and Section D 5 marks.
- Show all working for Sections B, C and D. Only final answers are given at the end — for full solutions, raise your doubts with your teacher.
Section A — Multiple Choice Questions
5 × 1 = 5 marks
1.
The area under $y=x^2$ from $0$ to $3$ is:
- A.$9$
- B.$27$
- C.$3$
- D.$6$
2.
Area between $y=x$ and $y=x^2$ on $[0,1]$ is:
- A.$\tfrac16$
- B.$\tfrac13$
- C.$\tfrac12$
- D.$1$
3.
$\displaystyle\int_0^{\pi}\sin x\,dx$ (area) equals:
- A.$0$
- B.$1$
- C.$2$
- D.$\pi$
4.
The limits for area between two curves come from:
- A.the $y$-intercepts
- B.their intersection points
- C.the origin
- D.the derivative
5.
If $f(x)<0$ on part of $[a,b]$, area is found using:
- A.the plain integral
- B.absolute values of the pieces
- C.the derivative
- D.half the integral
Section B — Short Answer (2 marks)
4 × 2 = 8 marks
6.
Find the area under $y=x^2$ from $x=0$ to $x=2$.
7.
Find the area between $y=x$ and $y=x^2$ from their intersections.
8.
Find the area under $y=2x$ from $x=0$ to $x=3$.
9.
Find the intersection points of $y=2x$ and $y=x^2$.
Section C — Short Answer (3 marks)
4 × 3 = 12 marks
10.
Find the area bounded by $y=x^2$, the $x$-axis and $x=3$.
11.
Find the area between $y=x^2$ and $y=4$.
12.
Find the area under $y=\sin x$ from $0$ to $\pi$.
13.
Find the area between $y=\sqrt{x}$ and $y=x$ for $0\le x\le 1$.
Section D — Long Answer (5 marks)
2 × 5 = 10 marks
14.
Find the area enclosed by the circle $x^2+y^2=4$.
15.
Find the area of the region bounded by the parabola $y^2=4x$ and the line $y=2x$.
Answer Key
Section A — Multiple Choice Questions
- (A) $9$
- (A) $\tfrac16$
- (C) $2$
- (B) their intersection points
- (B) absolute values of the pieces
Section B — Short Answer (2 marks)
- $\tfrac{8}{3}$ sq units.
- $\tfrac16$ sq unit.
- $9$ sq units.
- $(0,0)$ and $(2,4)$.
Section C — Short Answer (3 marks)
- $9$ sq units.
- $\tfrac{32}{3}$ sq units.
- $2$ sq units.
- $\tfrac16$ sq unit.
Section D — Long Answer (5 marks)
- $4\pi$ sq units.
- $\tfrac13$ sq unit.
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