Application of Integrals
Class 12 · Mathematics
4 ideas in Class 12 Mathematics, explained in plain words, with games and quizzes to practise.
Strips and Signs of Area
Part of: Strips, Signs and Symmetric Areas
Area grows by adding thin strips: vertical strips of height y and width dx, or horizontal strips of width x and height dy. Integrals below the axis come out negative, so area takes their absolute value, splitting mixed regions at axis crossings.
Why it matters. Every bounded region in the chapter is weighed strip by strip with correct signs.
Formulas
- Vertical area is integral of y dx
- Horizontal area is integral of x dy
- Split mixed regions at axis crossings
Key words
- Vertical strip
- : A thin slice of height y and width dx.
- Horizontal strip
- : A thin slice of width x and height dy.
- Elementary area
- : The small y dx or x dy piece being summed.
- Absolute area rule
- : Below-axis integrals contribute their positive size.
Try it in two minutes
- Draw thin vertical strips under a curve.
- Shade any part dipping below the axis.
- Split the integral where the curve crosses over.
Try at home. Weigh y = x² − 1 from 0 to 2 with a split at x = 1.
Symmetric Regions
Part of: Strips, Signs and Symmetric Areas
Circles and ellipses mirror across both axes, so one quadrant weighed and multiplied by 4 gives the whole. Regions hugging the y-axis integrate x with respect to y, turning lines into simple powers.
Why it matters. Symmetry quarters the work, and y-axis regions appear in every paper.
Formulas
- Circle radius r encloses πr²
- Ellipse axes a, b enclose πab
- One quadrant times 4 gives the whole
Key words
- Quadrant symmetry
- : Mirroring across both axes into four equal pieces.
- Circle area rule
- : Radius r encloses π times r squared.
- Ellipse area rule
- : Semi-axes a and b enclose π times a times b.
- y-axis region
- : A region weighed with horizontal strips in y.
Try it in two minutes
- Fold a circle sketch into four matching quarters.
- Weigh one quarter with vertical strips.
- Multiply by 4 for the whole.
Try at home. Weigh the first-quadrant piece of x² + y² = 36.
Areas Under Curves
Part of: Curves and Combined Regions
Parabolas integrate into cubic-over-three pieces, waves need hump-by-hump absolute weighing, and straight lines form trapeziums that integrals confirm. Any sign change along the way forces a split before adding.
Why it matters. These three curve families cover most area questions outside conics.
Formulas
- Parabola pieces weigh x³ over 3
- Waves need hump-wise absolute sums
- Lines weigh average height times width
Key words
- Ordinate
- : A vertical fence line like x = a.
- Hump
- : One above-axis lobe of a wave.
- Trapezium area
- : Average height times width for straight lines.
- Sign split
- : Breaking integrals where the curve crosses the axis.
Try it in two minutes
- Name the curve family first.
- Mark zeros inside the interval.
- Integrate piece by piece with absolute sizes.
Try at home. Weigh y = x² from 0 to 4 and compare with geometry.
Regions Between Curves
Part of: Curves and Combined Regions
Between two curves the upper minus the lower integrates strip by strip, starting from their meeting points. Mixed regions crossing the axis still need absolute pieces, and blindly integrating across a dip reports zero instead of area.
Why it matters. Enclosed regions and sign traps decide the hardest area marks.
Formulas
- Between curves integrate upper minus lower
- Start and end at meeting points
- Blind integrals across dips report zero
Key words
- Upper curve
- : The higher graph bounding a strip from above.
- Meeting point
- : An intersection starting or ending a region.
- Enclosed region
- : A park fenced fully by two or more curves.
- Sign trap
- : A dip that zeroes a blind integral.
Try it in two minutes
- Find where the curves meet.
- Name the upper and lower curves.
- Integrate upper minus lower between meetings.
Try at home. Weigh the region between y = x and y = x² from 0 to 1.
What your child will practise
- Areas Under Curves: Candy Puzzle (Clue Puzzle)
- Areas Under Curves: Ocean Quick Fire (Quick Fire)
- Regions Between Curves: Desert Puzzle (Clue Puzzle)
- Strips and Signs of Area: Candy Quick Fire (Quick Fire)
- Symmetric Regions: Ocean Quest (Flip and Match)
- Quiz: Areas Under Curves (8 questions)
- Quiz: Regions Between Curves (8 questions)
- Quiz: Strips and Signs of Area (8 questions)
- Quiz: Symmetric Regions (8 questions)