Mathematics · Textbook solutions

Application of Integrals

Every solved example, exercise, and miscellaneous question, in the order the textbook teaches them. Plus every board question asked from this chapter. · 14 questions

8.2 Area under Simple Curves

6 q

Solved Examples

Worked · 2

8.1 Eg.1 $x^2 + y^2 = a^2$
8.1 Eg.2 $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$

Exercise 8.1

Practice · 4

Ex 8.1 Q1 $\frac{x^2}{16} + \frac{y^2}{9} = 1$
Ex 8.1 Q2 $\frac{x^2}{4} + \frac{y^2}{9} = 1$
Ex 8.1 Q3 $x^2 + y^2 = 4$
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Ex 8.1 Q4 $y^2 = 4x$
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Miscellaneous Exercise on Chapter 8

8 q

Solved Examples

Worked · 2

Misc Eg.3 $y = 3x + 2$
Misc Eg.4 $y = \cos x$

Miscellaneous Exercise

Practice · 6

Find the area under the given curves and given lines:
Misc Q1(i) $y = x^2$
Misc Q1(ii) $y = x^4$
Misc Q2 $y = |x + 3|$
Misc Q3 $y = \sin x$
Misc Q4 $y = x^3$
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Misc Q5 $y = x|x|$
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What the board has asked from this chapter

Questions per paper · 26 papers · 2022 to 2026

Bars average the full papers of each year. 25 further questions come from set variants and are listed below.

2026

22 q

10 $y = x\left| x \right|$
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Area Under Simple Curves
10 $y = x^2$
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Area Bounded by a Curve and a Line
11 $y = x$
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Area Under Simple Curves
12 $The area of the shaded region of the circle given below is equal to :$
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Area Bounded by a Curve and a Line
12 $y = 2x - 4$
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Area Bounded by a Curve and a Line
33 $y = x|x|$
Area Under Simple Curves
33 $y = |x - 6|$
Area Under Simple Curves
33 $y = 5x + 4$
Area Under Simple Curves
34 $\{(x,\ y) : 9x^2 + 16y^2 = 144\}$
Area Under Simple Curves
34 $\{(x,\ y) : 100x^2 + 25y^2 = 2500\}$
Area Under Simple Curves
34 $\left\{(x,\ y) : \dfrac{x^2}{25} + \dfrac{y^2}{25} = 1\right\}$
Area Bounded by a Curve and a Line
A racing track is build around an elliptical ground whose equation is given by $9x^2 + 16y^2 = 144$ . The width of the track is 3 m as shown below : Based on given information, answer the following questions :
37 (i) $x$
Area Under Simple Curves
A racing track is build around an elliptical ground whose equation is given by $9x^2 + 16y^2 = 144$ . The width of the track is 3 m as shown below : Based on given information, answer the following questions :
37 (iii) (a) $Find the area of the region enclosed within the elliptical ground excluding the track using integration.$
Area Under Simple Curves
A racing track is build around an elliptical ground whose equation is given by $9x^2 + 16y^2 = 144$ . The width of the track is 3 m as shown below : Based on given information, answer the following questions :
37 (iii) (b) $x$
Area Bounded by a Curve and a Line
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 :
38 (i) $C_1$
Area Under Simple Curves
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 :
38 (i) $Write the equation of the boundary line AB of the park.$
Area Bounded by a Curve and a Line
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 :
38 (ii) $x$
Area Under Simple Curves
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 :
38 (ii) $Write the equation of the boundary line AC of the park.$
Area Bounded by a Curve and a Line
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 :
38 (iii) (a) $Using integration find the area of region covered by the roundabout.$
Area Under Simple Curves
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 :
38 (iii) (a) $Using integration, find the area of region OAC, in which children are allowed to play cricket, football.$
Area Bounded by a Curve and a Line
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 :
38 (iii) (b) $Using integration, find the area of region AOB.$
Area Bounded by a Curve and a Line
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 :
38 (iii) (b) $Using integration, find the area of region covered by circular pond.$
Area Under Simple Curves

2025

15 q

32 $y = \sqrt{x}$
Area Under Simple Curves
13 $y = x|x|$
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Area Under Simple Curves
15 $y = x^2$
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Area Under Simple Curves
17 $y^2 = x$
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Area Under Simple Curves
17 $y = \sqrt{x}$
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Area Under Simple Curves
17 $y^2 = x$
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Area Under Simple Curves
32 $y = 1 + |x + 1|$
Area Under Simple Curves
24 $\frac{x^2}{9} + \frac{y^2}{4} = 1$
Area Under Simple Curves
31 $y = |x + 3|$
Area Under Simple Curves
32 $y = 5x + 2$
Area Bounded by a Curve and a Line
32 $y = x^2$
Area Bounded by a Curve and a Line
33 $y = 2 + |x + 1|$
Area Under Simple Curves
33 $\dfrac{\pi}{4}$
Area Bounded by a Curve and a Line
33 $y = -x^2$
Area Under Simple Curves
33 $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$
Area Bounded by a Curve and a Line

2024

11 q

13 $y^2 = 4x$
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Area Under Simple Curves
13 $y^2 = 4x$
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Area Bounded by a Curve and a Line
13 $y = \sqrt{x}$
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Area Under Simple Curves
32 (a) $y = x\,|x|$
Area Under Simple Curves
32 (b) $9x^{2} + 25y^{2} = 225$
Area Bounded by a Curve and a Line
33 $x - 2y = 4$
Area Bounded by a Curve and a Line
33 $4x^2 + \mathrm{y}^2 = 36$
Area Under Simple Curves
34 $x^2 + y^2 = 16$
Area Bounded by a Curve and a Line
34 $\frac{x^2}{16} + \frac{y^2}{4} = 1$
Area Bounded by a Curve and a Line
35 $A_1$
Area Bounded by a Curve and a Line
35 $y = \sqrt{4 - x^2}$
Area Bounded by a Curve and a Line

2023

13 q

23 $2x + y = 8$
Area Bounded by a Curve and a Line
27 $y = mx$
Area Under Simple Curves
29 $x^2 + y^2 = 4$
Area Bounded by a Curve and a Line
29 $\{(x, y) : y^2 \le 2x \text{ and } y \ge x - 4\}$
Area Bounded by a Curve and a Line
32 $x^2 + y^2 = 16$
Area Bounded by a Curve and a Line
33 $x^2 + y^2 = 4$
Area Bounded by a Curve and a Line
33 $y^2 = 4ax$
Area Under Simple Curves
33 $-1$
Area Bounded by a Curve and a Line
33 $\{(x, y) : x^2 + y^2 \le 1 \le x + y\}$
Area Bounded by a Curve and a Line
34 $y = \sqrt{3}x$
Area Bounded by a Curve and a Line
34 $Find the area of the region bounded by the lines y = 4x + 5, x + y = 5 and 4y = x + 5, using integration.$
Area Bounded by a Curve and a Line
35 $x^2 = y$
Area Bounded by a Curve and a Line
35 $\frac{x^2}{25} + \frac{y^2}{16} = 1$
Area Bounded by a Curve and a Line

2022

13 q

7 $\{(x, y) : y^2 \le x \le y\}$
Area Bounded by a Curve and a Line
7 $\{(x, y) : x^2 \le y \le x + 2\}$
Area Bounded by a Curve and a Line
7 $y = mx$
Area Bounded by a Curve and a Line
11 (a) $\{(x, y) : 4x^2 + 9y^2 \leq 36, 2x + 3y \geq 6\}$
Area Bounded by a Curve and a Line
11 $4x^2 = y$
Area Bounded by a Curve and a Line
12 $y = |x - 1|$
Area Bounded by a Curve and a Line
11 (b) $x - y + 1 = 0$
Area Under Simple Curves
13 (a) $y = x^2$
Area Under Simple Curves
13 (a) $4x^2 + 4y^2 = 9$
Area Bounded by a Curve and a Line
12 $x^2 + y^2 = 4$
Area Bounded by a Curve and a Line
13 $y^2 = x$
Area Under Simple Curves
13 (b) $y^2 = 4ax$
Area Bounded by a Curve and a Line
13 (b) $y = \sqrt{3}x$
Area Bounded by a Curve and a Line

The 10 board questions to practise first

The questions the board asked again come first, most asked first. Each one says how often and when.

Question 1
Asked 3 times: 2024, 2025, 2026
33 $y = x|x|$
Question 2
Asked 2 times: 2024*, 2026 (* = numbers or wording changed that year)
34 $\{(x,\ y) : 9x^2 + 16y^2 = 144\}$
Question 3
Asked 2 times: 2023*, 2026 (* = numbers or wording changed that year)
34 $\left\{(x,\ y) : \dfrac{x^2}{25} + \dfrac{y^2}{25} = 1\right\}$
Question 4
Asked 2 times: 2025*, 2026 (* = numbers or wording changed that year)
33 $y = 5x + 4$
Question 5
Asked 2 times: 2024*, 2025 (* = numbers or wording changed that year)
33 $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$
Question 6
Asked 2 times: 2022*, 2023 (* = numbers or wording changed that year)
34 $y = \sqrt{3}x$
Question 7
Asked 2 times: 2022*, 2023 (* = numbers or wording changed that year)
35 $x^2 = y$
Question 8
Asked 2 times: 2022*, 2023 (* = numbers or wording changed that year)
35 $\frac{x^2}{25} + \frac{y^2}{16} = 1$
Question 9
Asked 2 times: 2022*, 2023 (* = numbers or wording changed that year)
33 $\{(x, y) : x^2 + y^2 \le 1 \le x + y\}$
Question 10
This type asked 5 times: 2022, 2023, 2024, 2025, 2026. This one: 2026
12 $y = 2x - 4$
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8.2 Area under Simple Curves

Solved Examples

Exercise 8.1

Miscellaneous Exercise on Chapter 8

Solved Examples

Miscellaneous Exercise

What the board has asked from this chapter

2026

2025

2024

2023

2022

The 10 board questions to practise first

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10