Class 12 Maths · Chapter 8
Unit: Calculus
Tentative Study Share: 7.0 Marks
Chapter 8: Applications Of Integrals — NCERT Solutions, Notes, Formula Sheet & PYQs
Complete NCERT exercise solutions, revision notes, formula cards, printable mind map, and solved board questions for Class 12 Applications Of Integrals.
NCERT Exercises in Chapter 8: Applications Of Integrals
📘 NCERT Concept Notes — Applications Of Integrals
01 · Area is built from strips
Sketch the region before integrating. A vertical strip has width dx and height measured above the x-axis. A horizontal strip has thickness dy and width measured from the y-axis. Choose the variable that gives the simpler description.
02 · Geometric area does not cancel
A signed integral can be negative or zero even when the region has positive area. Split wherever the curve changes sign, and make each contribution nonnegative.
03 · Symmetry can reduce the work
Use only a symmetry that belongs to both the curve and the chosen region. An ellipse has four congruent quadrant regions, so integrate one quadrant and multiply by four.
⚡ Key Formulas & Results — Applications Of Integrals
Vertical strips
How do you calculate area under a nonnegative curve?
\(A=\int_a^bf(x)dx\)
Horizontal strips
What changes when the curve is written x=g(y)?
\(A=\int_c^dg(y)dy\)
Curve below the axis
What is the area when f is nonpositive?
\(A=-\int_a^bf(x)dx\)
Sign-changing curves
How do you prevent positive and negative regions from cancelling?
\(A=\int_a^b|f(x)|dx\)
Derive the area over one sine period
Why is the area from 0 to 2π equal to 4 rather than zero?
\(\int_0^{2\pi}|\sin x|dx=4\)
Derive the ellipse area
How does a quarter-ellipse integral become πab?
\(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\quad\Rightarrow\quad A=\pi ab\)
🎯 CBSE 2026 Board PYQs from Applications Of Integrals
CBSE Class 12 Mathematics 2026 Main Paper 65/1/1 · Q10
The area bounded by the curve y = x|x|, x-axis and the ordinates x = −1 and x = 1 is given by
CBSE Class 12 Mathematics 2026 Main Paper 65/1/1 · Q38
Roundabouts are often made on busy roads to ease the traffic and avoid red lights. One such roundabout…
CBSE Class 12 Mathematics 2026 Main Paper 65/1/2 · Q3
The area bounded by the curve y = x|x|, x-axis and the ordinates x = −1 and x = 1 is given by
CBSE Class 12 Mathematics 2026 Main Paper 65/1/2 · Q37
Roundabouts are often made on busy roads to ease the traffic and avoid red lights. One such roundabout…
CBSE Class 12 Mathematics 2026 Main Paper 65/1/3 · Q14
The area bounded by the curve y = x|x|, x-axis and the ordinates x = −1 and x = 1 is given by
CBSE Class 12 Mathematics 2026 Main Paper 65/1/3 · Q36
Roundabouts are often made on busy roads to ease the traffic and avoid red lights. One such roundabout…
CBSE Class 12 Mathematics 2026 Main Paper 65/2/1 · Q10
Which expression gives the area bounded by y=x² and y=16?
CBSE Class 12 Mathematics 2026 Main Paper 65/2/1 · Q37
A 3 m wide track surrounds an elliptical ground given by 9x²+16y²=144. The outer edge meets the…
CBSE Class 12 Mathematics 2026 Main Paper 65/2/2 · Q3
Which expression gives the area bounded by y=x² and y=16?
CBSE Class 12 Mathematics 2026 Main Paper 65/2/2 · Q36
A 3 m wide track surrounds an elliptical ground given by 9x²+16y²=144. The outer edge meets the…
CBSE Class 12 Mathematics 2026 Main Paper 65/2/3 · Q14
Which expression gives the area bounded by y=x² and y=16?
CBSE Class 12 Mathematics 2026 Main Paper 65/2/3 · Q38
A 3 m wide track surrounds an elliptical ground given by 9x²+16y²=144. The outer edge meets the…
CBSE Class 12 Mathematics 2026 Main Paper 65/3/1 · Q12
Which integral gives the area of the shaded region in the circle diagram?
CBSE Class 12 Mathematics 2026 Main Paper 65/3/1 · Q38
A triangular park has vertices A=(0,4), B=(−2,0) and C=(3,0). O is the origin. Segment AO divides the…
CBSE Class 12 Mathematics 2026 Main Paper 65/3/2 · Q5
Which integral gives the area of the shaded region in the circle diagram?
CBSE Class 12 Mathematics 2026 Main Paper 65/3/2 · Q37
A triangular park has vertices A=(0,4), B=(−2,0) and C=(3,0). O is the origin. Segment AO divides the…
CBSE Class 12 Mathematics 2026 Main Paper 65/3/3 · Q16
Which integral gives the area of the shaded region in the circle diagram?
CBSE Class 12 Mathematics 2026 Main Paper 65/3/3 · Q36
A triangular park has vertices A=(0,4), B=(−2,0) and C=(3,0). O is the origin. Segment AO divides the…
📝 Chapterwise Practice Questions — Applications Of Integrals (52 Questions)
Class 12 Applications Of Integrals · Assertion Reason
Assertion: The area between y=sin²x and the x-axis on [0,π] is greater than that for y=sin x on [0,π]…
Class 12 Applications Of Integrals · Assertion Reason
Assertion: The area inside 2x²+3y²=6 is greater than the area inside x²+y²−2x+4y+4=0. Reason: The…
Class 12 Applications Of Integrals · Assertion Reason
Assertion: The area enclosed by x²+y²=36 equals 36π square units. Reason: A circle x²+y²=r², r>0…
Class 12 Applications Of Integrals · Assertion Reason
Assertion: The first-quadrant region enclosed by y=x³ and y=x has area 1/4 square unit. Reason: A…
Class 12 Applications Of Integrals · Assertion Reason
Assertion: The region between x=y² and x=3 has area 4√3 square units. Reason: Intersections of y²=4ax…
Class 12 Applications Of Integrals · Case Study
An external elevator has an upper-semicircular floor. Its curved boundary is the upper half of x²+y²=4…
Class 12 Applications Of Integrals · Case Study
A simplified honeybee-nest profile is bounded by the parabola x²=4y and the horizontal branch line y=4…
Class 12 Applications Of Integrals · Case Study
A triangular field has edges y=2x−2, y=(x−1)/2 and y=4−x. For part (i), shade only the portion with…
Class 12 Applications Of Integrals · Short Answer
Find the area bounded by y=x², the x-axis, and x=−1,x=1.
Class 12 Applications Of Integrals · Short Answer
Sketch the region between y=√(4−x²) and the x-axis, and find its area by integration.
Class 12 Applications Of Integrals · Short Answer
For a>0, find the area under y=√(a²−x²), above the x-axis, between x=0 and x=a.
Class 12 Applications Of Integrals · Short Answer
Find the geometric area between y=sin x and the x-axis from x=0 to x=2π.
Class 12 Applications Of Integrals · Short Answer
Sketch the region bounded by 2x+y=8, y=2,y=4 and the y-axis. Find its area using integration.
Class 12 Applications Of Integrals · Short Answer
Find by integration the area enclosed by y=|x−1| and y=1.
Class 12 Applications Of Integrals · Very Long Answer
In the first quadrant, the region inside x²+y²=4, above the x-axis and below the ray y=mx (m>0) has…
Class 12 Applications Of Integrals · Very Long Answer
Using integration, find the area of the small region bounded by y=x², y=x+2 and the x-axis with −2≤x≤0…
Class 12 Applications Of Integrals · Very Long Answer
Using integration, find the first-quadrant area bounded by y=√3x, y=√(4−x²) and the y-axis.
Class 12 Applications Of Integrals · Very Long Answer
Find using integration the area of {(x,y): y²≤2x and y≥x−4}.
Class 12 Applications Of Integrals · Very Long Answer
Sketch and find by integration the area of {(x,y):0≤y≤x²,0≤y≤x,0≤x≤2}.
Class 12 Applications Of Integrals · Very Long Answer
Find the first-quadrant area inside x²+y²=4, above the x-axis and below y=√3x, using integration.
Class 12 Applications Of Integrals · Very Long Answer
Using integration, find the area of {(x,y):x²+y²≤9,x+y≥3}.
Class 12 Applications Of Integrals · Very Long Answer
Using integration, find the area of the triangle with vertices (2,-2), (4,5), (6,2).
Class 12 Applications Of Integrals · Very Long Answer
Using integration, find the area bounded by 3x−2y+1=0, 2x+3y−21=0 and x−5y+9=0.
Class 12 Applications Of Integrals · Very Long Answer
Find by integration the area of {(x,y):x²+y²≤4,x+y≥2}.