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The space occupied by the curve along with the axis, under the given condition is called area of bounded region.

(i) The area bounded by the curve y = F(x) above the X-axis and between the lines x = a, x = b is given by

(ii) If the curve between the lines x = a, x = b lies below the X-axis, then the required area is given by

(iii) The area bounded by the curve x = F(y) right to the Y-axis and the lines y = c, y = d is given by

(iv) If the curve between the lines y = c, y = d left to the Y-axis, then the area is given by

(v) Area bounded by two curves y = F (x) and y = G (x) between x = a and x = b is given by

(vi) Area bounded by two curves x = f(y) and x = g(y) between y=c and y=d is given by

(vii) If F (x) â‰¥. G (x) in [a, c] and F (x) â‰¤ G (x) in [c,d], where a < c < b, then area of the region bounded by the curves is given as

**Area of Curves Given by Polar Equations**

Let f(Î¸) be a continuous function, Î¸ âˆˆ (a, Î±), then the are t bounded by the curve r = f(Î¸) and radius Î±, Î²(Î± < Î²) is

**Area of Parametric Curves**

Let x = Ï†(t) and y = Ïˆ(t) be two parametric curves, then area bounded by the curve, X-axis and ordinates x = Ï†(t_{1}), x = Ïˆ(t_{2}) is

**Volume and Surface Area**

If We revolve any plane curve along any line, then solid so generated is called solid of revolution.

**1. Volume of Solid Revolution**

1. The volume of the solid generated by revolution of the area bounded by the curve y = f(x), the axis of x and the ordinates

it being given that f(x) is a continuous a function in the interval (a, b).

2. The volume of the solid generated by revolution of the area bounded by the curve x = g(y), the axis of y and two abscissas y = c and y = d is

it being given that g(y) is a continuous function in the interval (c, d).

**Surface of Solid Revolution**

(i) The surface of the solid generated by revolution of the area bounded by the curve y = f(x), the axis of x and the ordinates

is a continuous function in the interval (a, b).

(ii) The surface of the solid generated by revolution of the area bounded by the curve x = f (y), the axis of y and y = c, y = d is

continuous function in the interval (c, d).

**Curve Sketching**

**1. symmetry**

- If powers of y in a equation of curve are all even, then curve is symmetrical about X-axis.
- If powers of x in a equation of curve are all even, then curve is symmetrical about Y-axis.
- When x is replaced by -x and y is replaced by -y, then curve is symmetrical in opposite quadrant.
- If x and y are interchanged and equation of curve remains unchanged curve is symmetrical about line y = x.

**2. Nature of Origin**

- If point (0, 0) satisfies the equation, then curve passes through origin.
- If curve passes through origin, then equate low st degree term to zero and get equation of tangent. If there are two tangents, then origin is a double point.

**3. Point of Intersection with Axes**

- Put y = 0 and get intersection with X-axis, put x = 0 and get intersection with Y-axis.
- Now, find equation of tangent at this point i. e. , shift origin to the point of intersection and equate the lowest degree term to zero.
- Find regions where curve does not exists. i. e., curve will not exit for those values of variable when makes the other imaginary or not defined.

**4. Asymptotes**

- Equate coefficient of highest power of x and get asymptote parallel to X-axis.
- Similarly equate coefficient of highest power of y and get asymptote parallel to Y-axis.

**5. The Sign of (dy/dx)**

Find points at which (dy/dx) vanishes or becomes infinite. It gives us the points where tangent is parallel or perpendicular to the X-axis.

**6. Points of Inflexion**

and solve the resulting equation.If some point of inflexion is there, then locate it exactly.

Taking in consideration of all above information, we draw an approximate shape of the curve.

**Shape of Some Curves is Given Below**

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