✅ Area Under The Curve Formulas ⭐️⭐️⭐️⭐️⭐

1. Area bounded by a curve

(i) The area bounded by a Cartesian curve y = f(x), x-axis and abscissa x = a and x = b is given by,

(iii) If the equation of a curve is in parametric form, say x = f(t), y = g(t), then the area = 

where t₁ and t₂ are the values of t respectively corresponding to the values of a & b of x.

(iv) If the curve be symmetrical and suppose it has n symmetrical portions then total area = n × area of one symmetrical portion

(v) If some part of the curve lies below x-axis then its area is negative then area must be calculated separately using modules sign. For example

2. Symmetrical area

If the curve is symmetrical about a coordinate axis (or a line or origin), then we find the area of one symmetrical portion and multiply it by the number of symmetrical portions to get the required area.

3. Area between two curves

(i) When two curves intersect at two points and their common area lies between these points. If y = f₁(x) and y = f₂(x) are two curves where f₁(x) > f₂(x) which intersect at two points A (x = a) and B (x = b) and their common area lies between A & B, then their

(ii) When two curves intersect at a point and the area between them is bounded by x-axis. If y = f₁(x) and y = f₂(x) are two curves which intersect at P(α, β) & meet x-axis at A(a, 0), B(b, 0) respectively, then area between them and x- axis is given by

✅ 12th grade math formulas ⭐️⭐️⭐️⭐️⭐