Application of Integrals Class 12 Notes

• To find the area between f and g, first find where the curves intersect — those points are the limits of integration.
• The strip height is the difference of the functions at each x.
• Region splits into pieces whenever the curves cross inside the interval — integrate each piece separately.

• Area between a parabola
• $y^2=4ax$
• and its latus rectum
• $x=a$
• is
• $\frac{8a^2}{3}$
• .
• Area bounded by
• $y=x^2$
• and the line
• $y=x$
• is found from their intersection points
• $(0,0)$
• and
• $(1,1)$
• .
• A full ellipse
• $x^2/a^2+y^2/b^2=1$
• has area
• $\pi ab$
• ; a circle has area
• $\pi r^2$
• .

• Integrate along x by vertical strips when each region is bounded above and below by single curves.
• Integrate along y by horizontal strips when the curves are easier written as x = g(y).
• For
• $y^2=4ax$
• type curves, horizontal strips often avoid square roots.

Example: Find the area between $y=x^2$ and $y=x$.

Solution: Intersections at x = 0 and x = 1, with $x\ge x^2$ on [0,1]. Area $={\int }_0^1(x-x^2)dx={\left[\frac{x^2}{2}-\frac{x^3}{3}\right]}_0^1=\frac{1}{2}-\frac{1}{3}=\frac{1}{6}$.

Example: Find the area of the ellipse $x^2/9+y^2/4=1$.

Solution: $a=3$, $b=2$, so the area is $\pi ab=\pi (3)(2)=6\pi$ — four times one quadrant computed by a single integration.

Exercises (NCERT Q1–Q19)

• 1) Find the area of the region bounded by the parabola and the line
• 2) Find the area of the region bounded by , , and the x-axis in the first quadrant.
• 3) Find the area of the region bounded by the parabola , the y-axis and the lines and in the first quadrant.
• 4) Find the area of the region bounded by the ellipse
• 5) Find the area of the region bounded by the ellipse
• 6) Find the area of the region bounded by the curve from to
• 7) Find the area of the region bounded by the curve and the line
• 8) Find the area of the region in the first quadrant enclosed by the x-axis, the line and the circle
• 9) Using integration, find the area of the region bounded by the triangle whose vertices are and
• 10) Find the area of the region bounded by the curve and the lines , and the x-axis.
• 11) Find the area of the region bounded by the curve and the x-axis between and
• 12) Find the area of the region enclosed between the two circles and
• 13) Find the area of the region bounded by the curve , the x-axis and the lines and
• 14) Find the area of the circle which is interior to the parabola
• 15) Find the area bounded by the curves and the x-axis.
• 16) Find the area of the region bounded by the curve and the line
• 17) Find the area of the region enclosed between the two parabolas and
• 18) Find the area of the region enclosed between the circles and
• 19) The area bounded by the curve and the line is: (A) (B) (C) (D)

Answers at a glance

1. 1
   Q1 → answer not in source — verify against the textbook
2. 2
   Q2 → answer not in source — verify against the textbook
3. 3
   Q3 → answer not in source — verify against the textbook
4. 4
   Q4 → answer not in source — verify against the textbook
5. 5
   Q5 → answer not in source — verify against the textbook
6. 6
   Q6 → answer not in source — verify against the textbook
7. 7
   Q7 → answer not in source — verify against the textbook
8. 8
   Q8 → answer not in source — verify against the textbook
9. 9
   Q9 → answer not in source — verify against the textbook
10. 10
    Q10 → answer not in source — verify against the textbook
11. 11
    Q11 → answer not in source — verify against the textbook
12. 12
    Q12 → answer not in source — verify against the textbook
13. 13
    Q13 → answer not in source — verify against the textbook
14. 14
    Q14 → answer not in source — verify against the textbook
15. 15
    Q15 → answer not in source — verify against the textbook
16. 16
    Q16 → answer not in source — verify against the textbook
17. 17
    Q17 → answer not in source — verify against the textbook
18. 18
    Q18 → answer not in source — verify against the textbook
19. 19
    Q19 → Option (A) — 32/3.

MCQ Questions

1 Mark Questions

1. 1
   Write the integral that gives the area between the parabola y = x² and the x-axis from x = 0 to x = 2. (Ans: ∫₀² x² dx)
2. 2
   What does ∫ₐᵇ |f(x)| dx represent when f dips below the x-axis?
3. 3
   State the unit of area in such problems.
4. 4
   State the symmetry used to find the full area of y² = 4ax separated by the x-axis.

2 Mark Questions

1. 1
   Find the area bounded by y = x², the x-axis and the ordinates x = 1 and x = 2. (Ans: ∫₁² x² dx = 7/3 sq units)
2. 2
   Find the area of the curve y = sin x between x = 0 and x = π. (Ans: 2 sq units)
3. 3
   Find the area cut off by y = 2x and the x-axis between x = 0 and x = 3. (Ans: 9 sq units)

3 Mark Questions

1. 1
   Find the area of the region bounded by the parabola y² = 4x and the line x = 1. (Ans: ∫₀¹ 2·2√x dx = 8/3 sq units)
2. 2
   Find the area of the ellipse x²/a² + y²/b² = 1, using symmetry and the integral ∫√(a² − x²)dx. (Ans: πab)
3. 3
   Find the area between y = x and y = x². (Ans: intersection 0 and 1; area = ∫₀¹ (√y − y)dy or ∫₀¹(x − x²)dx = 1/6 sq units)

4 Mark Questions

1. 1
   Draw the region bounded by y = x² and the line y = x + 2, shade it, and find its area by integrating along y. Justify the limits from the intersection points. (Ans: area = 9/2 sq units)
2. 2
   Sketch the curves y = √x and y = x/2, find their intersection, and compute the area between them. (Ans: intersection at (0,0) and (4,2); area = ∫₀⁴(√x − x/2)dx = 4/3 sq units)