`1.` The area bounded by the continuous curve `x = f(y),` the axis of
y and the abscissa `y = a` and `y = b` (where `b > a`) is given by

`A = int_a^bf(y)dy = int_a^bxdy`

`2.` The area bounded by the straight line `y = a, y = b (a < b)` and
the curves `x = f(y)` and `x = g(y),` provided `f(y) < g(y)`
(where `a <= y <=b)`, is given by

`A = int_a^b[g(y) - f(y)]dy`

`3.` When two curves `x = f(y)` and `x = g(y)` intersect, the bounded
area is

`A = int_a^b[g(y) - f(y)]dy ;` where `a < b.`

where a and bare the roots of the equation `f(y) = g(y)`

`4.` If some part of a curve lies left toy-axis, then its area becomes
negative but area cannot be negative. Therefore, we take its
modulus.
If the curves crosses they-axis at `c`, then the area bounded by
the curve `x = f(y)` and abscissae `y = a` and `y = b`

(where `b > a`) is given by `A = |int_a^cf(y)dy| + |int_c^bf(y)dy|`

` A = int_a^cf(y)dy - int_c^bf(y)dy`

`5.` The area bounded by `x = f(y)` and `x = g(y)` (where `a < y < b`),
when they intersect at `y = c in (a, b)` is given by

`A = int_a^b |f(y) - g(y)| dy`

or `int_a^c(f(y)-g(y))dy+int_c^b(g(y)-f(y))dy`

`(1)` Area bounded by the curve `y^2 = 4ax ; x^2 = 4by` is equal to `(16 ab)/3;`

At point of intersection

`(x^2/(4b)) = 4ax => x^4 = 64 ab^2x`

` => x = 0, (64ab^2)^(1/3)`

Let `k = 4 (ab^2)^(1/3)`

`A = int_0^k(2sqrt a sqrt x-x^2/(4b))dx`

` = [2sqrta x^(3/2)/(3/2) - x^3/(12b)]_0^k = (4sqrta)/3 k^(3/2) - k^3/(12b) = 4/3sqrta 8(ab^2)^(1/2) - (64(ab^2))/(12b)`

` = (32)/3ab - (16)/3ab = (16ab)/3`

`(2)` Area bounded by the parabola `y^2 = 4ax` and `y = mx` is equal to `(8a^2)/(3m^2) :`

`y^2 = 4ax` and `y = mx`

`m^2x^2 = 4ax => x = 0, (4a)/m^2`

`Area = int_0^c (2sqrt a sqrtx - mx)dx` where `c = (4a)/m^2`

` = (2sqrta x^(3/2)/(3/2) - (mx^2)/2)_0^c = (4sqrta)/3c^(3/2) -(mc^2)/2`

` = (4sqrta)/3.(8asqrta)/m^3 - m/2.(16a^2)/m^4 = (32a^2)/(3m^3) - (8a^2)/m^3 = (8a^2)/(3m^3)`

`(3)` Area enclosed by `y^2 = 4ax` and its double ordinate at `x = a :`

(chord perpendicular to the axis of symmetry)
Required area `= OABO`

` = 2.int_0^a (2sqrt(ax))dx = 4sqrta (x^(3/2)/(3/2))_0^a`

` = 8/3 sqrta.(asqrta) = (8a^2)/3`

`(4)` Whole area of ellipse `x^2/a^2 + y^2/b^2 = 1` is equal to `pi ab`

`A = 4int_0^2 (bsqrt(1-x^2/a^2))dx`

Put `x = a sin theta`

`A = 4 int_0^(pi/r) ab cos^2 theta d theta = 4ab int_0^(pi/2) cos^2 theta d theta`

` = 4ab int_0^(pi/r) ((1+cos2theta)/2)d theta = 4ab(pi/4) = pi ab`

The approximate shape of a curve, the following procedure in order
SYMMETRY:
`(a)` Symmetry about `x-` axis

If the equation of the curve remain unchanged by replacing `y` by `- y` then the curve is symmetrical about
the x-axis.
`e.g., y^2 = 4ax.`

`(b)` Symmetry about `y-` axis

If the equation of the curve remain unchanged by replacing x by - x then the curve is symmetrical about
the `y-` axis.
`e.g., x^2 = 4ay`

`(c)` Symmetry about both axes

lf the equation of the curve remain unchanged by replacingx by `- x` and `y` by `- y` then the curve is
symmetrical about the axis of `'x'` as well as `'y'`.
`e.g., x^2 + y^2 = a^2`

`(d)` Symmetry about the line `y = x`

If the equation of curve remains unchanged on interchaning 'x' and 'y', then the curve is symmetrical
aboutthe line `y = x`
`e.g., x^3 + y^3 = 3xy.`

(II) Find the points where the curve crosses the `x-` axis and the `y-` axis.

(III) Find `(dy)/(dx)`

(IV) Examine `y` when `x -> oo` or `x -> oo .`