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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 Xaxis 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 Xaxis, then the required area is given by
(iii) The area bounded by the curve x = F(y) right to the Yaxis and the lines y = c, y = d is given by
(iv) If the curve between the lines y = c, y = d left to the Yaxis, 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) ≥. G (x) in [a, c] and F (x) ≤ 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(θ) be a continuous function, θ ∈ (a, α), then the are t bounded by the curve r = f(θ) and radius α, β(α < β) is
Area of Parametric Curves
Let x = φ(t) and y = ψ(t) be two parametric curves, then area bounded by the curve, Xaxis and ordinates x = φ(t_{1}), x = ψ(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
2. Nature of Origin
3. Point of Intersection with Axes
4. Asymptotes
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 Xaxis.
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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209 videos218 docs139 tests
