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Area bounded by Functions

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 

Area of Bounded Region | Mathematics (Maths) for JEE Main & Advanced

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

Area of Bounded Region | Mathematics (Maths) for JEE Main & Advanced

(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

Area of Bounded Region | Mathematics (Maths) for JEE Main & Advanced

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

Area of Bounded Region | Mathematics (Maths) for JEE Main & Advanced

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

Area of Bounded Region | Mathematics (Maths) for JEE Main & Advanced

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

Area of Bounded Region | Mathematics (Maths) for JEE Main & Advanced

(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 Bounded Region | Mathematics (Maths) for JEE Main & Advanced

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 Bounded Region | Mathematics (Maths) for JEE Main & Advanced

Area of Parametric Curves

Let x = φ(t) and y = ψ(t) be two parametric curves, then area bounded by the curve, X-axis and ordinates x = φ(t1), x = ψ(t2) is

Area of Bounded Region | Mathematics (Maths) for JEE Main & Advanced

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

          Area of Bounded Region | Mathematics (Maths) for JEE Main & Advanced

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

Area of Bounded Region | Mathematics (Maths) for JEE Main & Advanced

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

Area of Bounded Region | Mathematics (Maths) for JEE Main & Advanced

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

  Area of Bounded Region | Mathematics (Maths) for JEE Main & Advanced

continuous function in the interval (c, d).

Curve Sketching

1. Symmetry

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

2. Nature of Origin

  1. If point (0, 0) satisfies the equation, then curve passes through origin.
  2. 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

  1. Put y = 0 and get intersection with X-axis, put x = 0 and get intersection with Y-axis.
  2. 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.
  3. 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

  1. Equate coefficient of highest power of x and get asymptote parallel to X-axis.
  2. 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

Area of Bounded Region | Mathematics (Maths) for JEE Main & Advanced

 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

Area of Bounded Region | Mathematics (Maths) for JEE Main & Advanced

Area of Bounded Region | Mathematics (Maths) for JEE Main & Advanced

Area of Bounded Region | Mathematics (Maths) for JEE Main & Advanced

Area of Bounded Region | Mathematics (Maths) for JEE Main & Advanced

Area of Bounded Region | Mathematics (Maths) for JEE Main & Advanced

Area of Bounded Region | Mathematics (Maths) for JEE Main & Advanced

Area of Bounded Region | Mathematics (Maths) for JEE Main & Advanced

Area of Bounded Region | Mathematics (Maths) for JEE Main & Advanced

Area of Bounded Region | Mathematics (Maths) for JEE Main & Advanced

Area of Bounded Region | Mathematics (Maths) for JEE Main & Advanced

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FAQs on Area of Bounded Region - Mathematics (Maths) for JEE Main & Advanced

1. How do you find the area of curves given by polar equations?
Ans. To find the area of curves given by polar equations, you can use the formula A = 0.5∫[r(θ)]^2 dθ, where r(θ) represents the polar equation of the curve.
2. How do you calculate the area of parametric curves?
Ans. To calculate the area of parametric curves, you can use the formula A = ∫[y(t)x'(t)] dt, where x(t) and y(t) are the parametric equations defining the curve.
3. How can you determine the volume and surface area of a solid formed by rotating a curve around an axis?
Ans. To determine the volume of a solid formed by rotating a curve around an axis, you can use the method of disks or washers. The surface area can be calculated using the formula A = 2π∫[y(x)√(1 + [y'(x)]^2)] dx.
4. What techniques can be used for curve sketching to understand the behavior of a function?
Ans. Techniques such as finding intercepts, critical points, asymptotes, and analyzing the behavior at the ends of the domain can be used for curve sketching to understand the function's behavior.
5. How can the area of a bounded region be determined using integration techniques?
Ans. The area of a bounded region can be determined by integrating the difference of the upper and lower curves with respect to the variable of integration over the specified interval.
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