Application of Integrals — Class 12 CBSE Mathematics revision notes
A 4-page visual revision sheet for Application of Integrals, written for Class 12 CBSE Mathematics. Formulas, diagrams and the traps that cost marks, laid out to be read the night before rather than the term before. Free, and it opens inside CapCompass.
Page 1 as PDF
Application of Integrals in one look
a definite integral read as area, one boundary curve at a time
Area under a simple curve
- region bounded by y=f(x), the x-axis, and the lines x=a, x=b, with f(x)>=0 on [a,b]
Area = ∫_a^b y dx - if the curve crosses the x-axis at x=c inside [a,b], split there first
Area = Big|∫_a^c y dxBig|+Big|∫_c^b y dxBig|
Reading a definite integral as area
- a definite integral is the signed value the fundamental theorem returns, not automatically an area
- calling it an area needs the sign checked first; any negative part is made positive
- a shaded figure shows the exact region; an unshaded boundary curve still needs it imagined
Area bounded by a simple curve
Q. Find the area bounded by y=x^2, the x-axis, and the lines x=1 and x=3.
- Area = the integral of x^2 dx from x=1 to x=3
- antiderivative: x^3/3
- apply the limits: (3^3/3) - (1^3/3) = 27/3 - 1/3
Answer: 26/3 square units
Area bounded by a curve and the y-axis
- region bounded by x=f(y), the y-axis, and the lines y=c, y=d, with f(y)>=0 on [c,d]
Area = ∫_c^d x dy
Page 1 of 4. The full sheet, all 4 pages, opens free in the app.
What this Application of Integrals sheet covers
- Area under a simple curve
- region bounded by y=f(x), the x-axis, and the lines x=a, x=b, with f(x)>=0 on [a,b]
- if the curve crosses the x-axis at x=c inside [a,b], split there first
- Reading a definite integral as area
- Area bounded by a simple curve
- Area bounded by a curve and the y-axis
- region bounded by x=f(y), the y-axis, and the lines y=c, y=d, with f(y)>=0 on [c,d]
- A curve that closes on itself
- Circles and ellipses, from integration
- circle of radius r, centred at the origin
- ellipse with semi-axes a and b
- Area between two curves
- f(x)>=g(x) throughout [a,b]
- Which curve is on top, and do they cross
- Setting up the integral correctly
- Symmetry shortcuts
- curve symmetric about the y-axis, region spanning x=-a to x=a
- curve symmetric about the x-axis, region spanning y=-b to y=b
- Choosing the strip: dx vs dy
- dx strips (integrate in x)
- dy strips (integrate in y)
- Common enclosed shapes and their setup
- Circle
- Parabola and a line
- Ellipse
- Two intersecting parabolas
- Traps that cost marks
- Tricks that save time
How CapCompass uses this sheet
These notes are not a standalone PDF. CapCompass plans Mathematics chapter by chapter against your real exam dates and free hours, and when the plan reaches Application of Integrals the sheet opens inside the study block itself — at the page covering the sub-topic that block is on, with the timer still running. Revision arrives at the moment the plan says revise, instead of having to be gone looking for.