CBSE Maths › Class 12 PYQs › Application of Integrals

Application of Integrals: 4 marks Questions (CBSE Class 12)

3 different 4 marks questions on Application of Integrals from CBSE Class 12 Maths board exams 2026, newest first.

Q384 marksCase StudyApplication of Integrals CBSE 2026 · 65/1/1

Roundabouts are often made on busy roads to ease the traffic and avoid red lights.
One such round-about is made such that equation representing its boundary is given by $C_{1}; x^{2} + y^{2} = 64$ .
There is a circular pond with a fountain in the middle of the roundabout whose equation is given by $C_{2} : x^{2} + y^{2} = 4$ .
Based on the given information, answer the following questions :
(i) Represent the given equations $C_{1}$ and $C_{2}$ with the help of a diagram. (1)
(ii) Express y as a function of $x$ , (y = f(x)), for both $C_{1}$ an $C_{2}$ . (1)
(iii) Using integration find the area of region covered by the roundabout. (2)
OR (iii) Using integration, find the area of region covered by circular pond. (2)

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Answer: (i) Two concentric circles with centre O(0, 0) and radii 8 and 2 (ii)

C_{1}

y = \pm 64 - x^{2}

;

C_{2}

y = \pm 4 - x^{2}

(iii)

64 π

sq units; OR

4 π

sq units

(i) $C_{1}$ is a circle with centre (0, 0) and radius 8; $C_{2}$ is a circle with centre (0, 0) and radius 2, inside $C_{1}$ .
(ii) $C_{1}$ : $y = \pm 64 - x^{2}$ , $- 8 \leq x \leq 8$ ; $C_{2}$ : $y = \pm 4 - x^{2}$ , $- 2 \leq x \leq 2$ .
(iii) By symmetry, area $= 4 \int_{0}^{8} 64 - x^{2} d x = 4 [\frac{x}{2} 64 - x^{2} + \frac{64}{2} sin^{- 1} \frac{x}{8}]_{0}^{8} = 4 \times 32 \times \frac{π}{2} = 64 π$ sq units.
OR (iii) Area $= 4 \int_{0}^{2} 4 - x^{2} d x = 4 [\frac{x}{2} 4 - x^{2} + 2 sin^{- 1} \frac{x}{2}]_{0}^{2} = 4 \times 2 \times \frac{π}{2} = 4 π$ sq units.

Also asked in: 2026 65/1/2, 2026 65/1/3

Q374 marksCase StudyApplication of Integrals CBSE 2026 · 65/2/1

A racing track is build around an elliptical ground whose equation is given by $9 x^{2} + 16 y^{2} = 144$ . The width of the track is 3 m as shown below :
Based on given information, answer the following questions :
(i) Express y as a function of $x$ from the given equation of ellipse. (1)
(ii) Integrate the function obtained in (i) with respect to $x$ . (1)
(iii) (a) Find the area of the region enclosed within the elliptical ground excluding the track using integration. (2)
OR (iii) (b) Write the co-ordinates of the points P and Q where the outer edge of the track cuts $x$ axis and y axis in first quadrant and find the area of the triangle formed by points P, O, Q using integration. (2)

Diagram for CBSE 2026 Class 12 Maths question 37

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Answer: (i)

y = \pm \frac{3}{4} 16 - x^{2}

(ii)

\frac{3}{8} x 16 - x^{2} + 6 sin^{- 1} \frac{x}{4} + C

(iii)(a)

12 π

sq m OR (iii)(b) P(7, 0), Q(0, 6); area 21 sq m

(i) $16 y^{2} = 144 - 9 x^{2} \Rightarrow y = \pm \frac{3}{4} 16 - x^{2}$ (upper half: $y = \frac{3}{4} 16 - x^{2}$ ).
(ii) $\int \frac{3}{4} 16 - x^{2} d x = \frac{3}{4} [\frac{x}{2} 16 - x^{2} + 8 sin^{- 1} \frac{x}{4}] + C = \frac{3}{8} x 16 - x^{2} + 6 sin^{- 1} \frac{x}{4} + C$ .
(iii)(a) Area $= 4 \int_{0}^{4} \frac{3}{4} 16 - x^{2} d x = 3 [\frac{x}{2} 16 - x^{2} + 8 sin^{- 1} \frac{x}{4}]_{0}^{4} = 3 (8 \cdot \frac{π}{2}) = 12 π$ sq m.
(iii)(b) Outer edge has semi-axes $4 + 3 = 7$ and $3 + 3 = 6$ : P(7, 0), Q(0, 6).
Line PQ: $\frac{x}{7} + \frac{y}{6} = 1 \Rightarrow y = 6 - \frac{6 x}{7}$ .
Area of $△ P O Q = \int_{0}^{7} (6 - \frac{6 x}{7}) d x = 42 - 21 = 21$ sq m.

Also asked in: 2026 65/2/2, 2026 65/2/3

Q384 marksCase StudyApplication of Integrals CBSE 2026 · 65/3/1

There is a triangular park in the society. The park is divided into two sections as shown in the figure.
In the region OAC, children are allowed to play games like cricket, football, while in the region AOB, activities which involve running are not allowed.
The vertices of the triangular park ABC are A(0, 4), B(– 2, 0) and C(3, 0).
Based on the above information, answer the following questions :
(i) Write the equation of the boundary line AB of the park. (1)
(ii) Write the equation of the boundary line AC of the park. (1)
(iii) (a) Using integration, find the area of region OAC, in which children are allowed to play cricket, football. (2)
OR (iii) (b) Using integration, find the area of region AOB. (2)

Diagram for CBSE 2026 Class 12 Maths question 38

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Answer: (i)

y = 2 x + 4

(ii)

4 x + 3 y = 12

(iii)(a) 6 sq units (iii)(b) 4 sq units

(i) Slope of AB $= \frac{4 - 0}{0 + 2} = 2$ , so $y = 2 x + 4$ .
(ii) Slope of AC $= \frac{0 - 4}{3 - 0} = - \frac{4}{3}$ , so $y = 4 - \frac{4 x}{3}$ , i.e. $4 x + 3 y = 12$ .
(iii)(a) Area OAC $= \int_{0}^{3} (4 - \frac{4 x}{3}) d x = [4 x - \frac{2 x ^{2}}{3}]_{0}^{3} = 12 - 6 = 6$ sq units.
(iii)(b) Area AOB $= \int_{- 2}^{0} (2 x + 4) d x = [x^{2} + 4 x]_{- 2}^{0} = 0 - (4 - 8) = 4$ sq units.

Also asked in: 2026 65/3/2, 2026 65/3/3

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