Area Under The Curve Formulas

Have you ever tried solving the Area bounded by a curve, Area Between two curves, Symmetrical area problems using the Area Under the Curve Formulas? Then, this is the right time to do. This page contains a list of Area under the Curve Formulas. So, make the most out of the area under the curve formulae sheet and solve the problems easily. Also, get familiar with all fundamentals of the Area Under the Curve concept by referring to the formulas list prevailing here.

List of Area Under The Curve Formulas

Take a look at the below-provided Area Under The Curve Formulas Sheet & List & memorize all formulas efficiently. Use the handles existed here and make your Area Under The Curve calculations at a faster pace during exam preparation or homework.

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,
Area =

${\int }_{a}^{b}$

y dx =

${\int }_{a}^{b}$

f(x) dx (ii) The area bounded by a Cartesian curve x = f(y), y-axis and ordinates y = c and y = d
Area =

${\int }_{c}^{d}$

x dx =

${\int }_{c}^{d}$

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

${\int }_{a}^{b}$

y dx =

${\int }_{{t}_{1}}^{{t}_{2}}$

g(t)f'(t)dt
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 (vi)  In all above cases

${\int }_{c}^{b}$

(f(x) – g(x)) dx =

${\int }_{a}^{b}$

f(x) dx –

${\int }_{a}^{b}$

g(x) dx

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
Common area =

${\int }_{a}^{b}$

(y1 – y2) dx (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 Area =

${\int }_{a}^{\alpha }$

f1(x) dx +

${\int }_{\alpha }^{a}$

f2(x) dx

(i) Equation of ellipse

$\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1$ Area of ellipse = π ab

(ii) Area enclosed =

$\frac{16ab}{3}$ (iii) Area enclosed =

$\frac{8{a}^{2}}{3}$ (iv) Area enclosed =

$\frac{8{a}^{2}}{3{m}^{3}}$ 