Skip to main content

Area under the curve - Notes, Concept and All Important Formula

AREA UNDER THE CURVE

Area bounded by the curve
1. The area bounded by the curve \(y=f(x)\), the \(x\) -axis and the ordinates \(x=a \) & \(x=b\) is given by, 
\(A=\displaystyle \int_{a}^{b} f(x) \,\, dx=\displaystyle \int_{a}^{b} y \,\, dx, f(x) \geq 0\)

2. If the area is below the \(x\) -axis then \(A\) is negative. The convention is to consider the magnitude only i.e.
\(A=\left|\displaystyle \int_{a}^{b} y \,\, dx\right|\) in this case.

area bounded by curve taking small division of y axis.
3. The area bounded by the curve \(x = f ( y ),\) y -axis \(\&\) abscissa \(y = c\), \(y=d\) is given by, Area \(=\displaystyle \int_{c}^{d} x d y=\displaystyle \int_{c}^{d} f(y) d y, f(y) \geq 0\)

Area between two curve
4. Area between the curves \(y=f(x) \) & \(y=g(x)\) between the ordinates \(x=a\) & \( x=b\) is given by,
\(A=\displaystyle \int_{a}^{b} f(x) \,\, dx-\displaystyle \int_{a}^{b} g(x) \,\, dx\)
\(=\displaystyle \int_{a}^{b}[f(x)-g(x)] \,\, dx, f(x) \geq g(x) \,\, \forall x \in(a, b)\)

5. Average value of a function \(y=f(x)\) w.r.t. \(x\) over an interval \(a \leq x \leq\) \(b\) is defined as: \( y(a v)=\dfrac{1}{b-a} \displaystyle \int_{a}^{b} f(x) \,\, dx\).


6. CURVE TRACING :

The following outline procedure is to be applied in Sketching the graph of a function \(y=f(x)\) which in turn will be extremely useful to quickly and correctly evaluate the area under the curves.

(a) Symmetry : The symmetry of the curve is judged as follows:

(i) If all the powers of \(y\) in the equation are even then the curve is symmetrical about the axis of \(x\).

(ii) If all the powers of \(x\) are even, the curve is symmetrical about the axis of \(y\).

(iii) If powers of \(x \) & \(y\) both are even, the curve is symmetrical about the axis of \(x\) as well as \(y\).

(iv) If the equation of the curve remains unchanged on interchanging \(x\) and \(y\), then the curve is symmetrical about \(y=x\)

(v) If on replacing 'x' by '-x' and 'y' by '- \(y\) ', the equation of the curve is unaltered then there is symmetry in opposite quadrants, i.e. symmetric about the origin.


(b) Find \(\dfrac{dy }{ dx}\)  & equate it to zero to find the points on the curve where you have horizontal tangents.

(c) Find the points where the curve crosses the \(x\) -axis & also the \(y\) -axis.

(d) Examine if possible the intervals when \(f ( x )\) is increasing or decreasing. Examine what happens to 'y' when \(x \rightarrow \infty\) or \(-\infty\).


7. USEFUL RESULTS :

(a) Whole area of the ellipse, \(\dfrac{x ^{2} }{ a ^{2}}+\dfrac{ y ^{2} }{ b ^{2}}=1\) is \(\pi ab\).

(b) Area enclosed between the parabolas \(y^{2}=4 a x \) & \(x^{2}=4by\) is \(16 ab / 3\)

(c) Area included between the parabola \(y^{2}=4\)ax & the line \(y=m x\) is \(8 a^{2} / 3 m^{3}\)

(d) Area enclosed by the parabola and its double ordinate \(P , Q\) is two-third of area of rectangle PQRS, where R,S lie on tangent at the vertex.

Labels:

Comments

Post a Comment

Popular posts from this blog

Indefinite Integration - Notes, Concept and All Important Formula

INDEFINITE INTEGRATION If  f & F are function of \(x\) such that \(F^{\prime}(x)\) \(=f(x)\) then the function \(F\) is called a PRIMITIVE OR ANTIDERIVATIVE OR INTEGRAL of \(\mathrm{f}(\mathrm{x})\) w.r.t. \(\mathrm{x}\) and is written symbolically as \(\displaystyle \int \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x}\) \(=\mathrm{F}(\mathrm{x})+\mathrm{c} \Leftrightarrow \dfrac{\mathrm{d}}{\mathrm{dx}}\{\mathrm{F}(\mathrm{x})+\mathrm{c}\}\) \(=\mathrm{f}(\mathrm{x})\) , where \(\mathrm{c}\) is called the constant of integration. Note : If \(\displaystyle \int f(x) d x\) \(=F(x)+c\) , then \(\displaystyle \int f(a x+b) d x\) \(=\dfrac{F(a x+b)}{a}+c, a \neq 0\) All Chapter Notes, Concept and Important Formula 1. STANDARD RESULTS : (i) \( \displaystyle \int(a x+b)^{n} d x\) \(=\dfrac{(a x+b)^{n+1}}{a(n+1)}+c ; n \neq-1\) (ii) \(\displaystyle \int \dfrac{d x}{a x+b}\) \(=\dfrac{1}{a} \ln|a x+b|+c\) (iii) \(\displaystyle \int e^{\mathrm{ax}+b} \mathrm{dx}\) \(=\dfrac{1}{...
Post a Comment
Read more

3d Coordinate Geometry - Notes, Concept and All Important Formula

3D-COORDINATE GEOMETRY 1. DISTANCE FORMULA: The distance between two points \(A \left( x _{1}, y _{1}, z _{1}\right)\) and \(B \left( x _{2}, y _{2}, z _{2}\right)\) is given by \(A B=\sqrt{\left[\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}+\left(z_{2}-z_{1}\right)^{2}\right]}\) All Chapter Notes, Concept and Important Formula 2. SECTION FORMULAE : Let \(P \left( x _{1}, y _{1}, z _{1}\right)\) and \(Q \left( x _{2}, y _{2}, z _{2}\right)\) be two points and let \(R ( x , y , z )\) divide \(PQ\) in the ratio \(m _{1}: m _{2}\) . Then \(R\) is \((x, y, z)=\left(\frac{m_{1} x_{2}+m_{2} x_{1}}{m_{1}+m_{2}}, \frac{m_{1} y_{2}+m_{2} y_{1}}{m_{1}+m_{2}}, \frac{m_{1} z_{2}+m_{2} z_{1}}{m_{1}+m_{2}}\right)\) If \(\left( m _{1} / m _{2}\right)\) is positive, \(R\) divides \(PQ\) internally and if \(\left( m _{1} / m _{2}\right)\) is negative, then externally. Mid point of \(PQ\) is given by \(\left(\frac{ x _{1}+ x _{2}}{2}, \frac{ y _{1}+ y _{2}}{2}, \frac{ z _{1}+ z ...
Post a Comment
Read more

Parabola - Notes, Concept and All Important Formula

PARABOLA 1. CONIC SECTIONS : A conic section, or conic is the locus of a point which moves in a plane so that its distance from a fixed point is in a constant ratio to its perpendicular distance from a fixed straight line. (a) The fixed point is called the FOCUS. (b) The fixed straight line is called the DIRECTRIX. (c) The constant ratio is called the ECCENTRICITY denoted by e. (d) The line passing through the focus & perpendicular to the directrix is called the AXIS. (e) A point of intersection of a conic with its axis is called a VERTEX. All Chapter Notes, Concept and Important Formula 2. GENERAL EQUATION OF A CONIC : FOCAL DIRECTRIX PROPERTY: The general equation of a conic with focus \((p, q) \) & directrix \(\mathrm{lx}+\mathrm{my}+\mathrm{n}=0\)   is :  \(\left(1^{2}+\mathrm{m}^{2}\right)\left[(x-\mathrm{p})^{2}+(\mathrm{y}-\mathrm{q})^{2}\right] \) \(=e^{2}(\mathrm{l} \mathrm{x}+\mathrm{m} y+\mathrm{n})^{2}  \) \(\equiv \mathrm{ax}^{2}+2 \mathrm{~h} ...
Post a Comment
Read more