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

Tangent & Normal - Notes, Concept and All Important Formula

TANGENT & NORMAL 1. TANGENT TO THE CURVE AT A POINT: The tangent to the curve at 'P' is the line through P whose slope is limit of the secant's slope as \(Q \rightarrow P\) from either side. All Chapter Notes, Concept and Important Formula 2. NORMAL TO THE CURVE AT A POINT: A line which is perpendicular to the tangent at the point of contact is called normal to the curve at that point. 3. THINGS TO REMEMBER : (a) The value of the derivative at \(\mathrm{P}\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right)\) gives the slope of the tangent to the curve at P. Symbolically \(\left.\mathrm{f}^{\prime}\left(\mathrm{x}_{1}\right)=\dfrac{\mathrm{dy}}{\mathrm{dx}}\right]_{\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right)}=\) Slope of tangent at \(\mathrm{P}\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right)=\mathrm{m}\) (say). (b) Equation of tangent at \(\left(x_{1}, y_{1}\right)\) is \(\left.y-y_{1}=\dfrac{d y}{d x}\right]_{\left(x_{1}, y_{1}\right)}\left(x-x_{1}\right)\) (c) Equation of...
Post a Comment
Read more

Trigonometry Ratios and Identities - Notes, Concept and All Important Formula

TRIGONOMETRIC RATIOS & IDENTITIES Table Of Contents 1. RELATION BETWEEN SYSTEM OF MEASUREMENT OF ANGLES : \(\dfrac{D}{90}=\dfrac{G}{100}=\dfrac{2 C}{\pi}\) 1 Radian \(=\dfrac{180}{\pi}\) degree \(\approx 57^{\circ} 17^{\prime} 15^{\prime \prime}\) (approximately) 1 degree \(=\dfrac{\pi}{180}\) radian \(\approx 0.0175\) radian All Chapter Notes, Concept and Important Formula 2. BASIC TRIGONOMETRIC IDENTITIES : (a) \(\sin ^{2} \theta+\cos ^{2} \theta=1\) or \(\sin ^{2} \theta=1-\cos ^{2} \theta\) or \(\cos ^{2} \theta=1-\sin ^{2} \theta\) (b) \(\sec ^{2} \theta-\tan ^{2} \theta=1\) or \(\sec ^{2} \theta=1+\tan ^{2} \theta\) or \(\tan ^{2} \theta=\sec ^{2} \theta-1\) (c) If \(\sec \theta+\tan \theta\) \(=\mathrm{k} \Rightarrow \sec \theta-\tan \theta\) \(=\dfrac{1}{\mathrm{k}} \Rightarrow 2 \sec \theta\) \(=\mathrm{k}+\dfrac{1}{\mathrm{k}}\) (d) \(\operatorname{cosec}^{2} \theta-\cot ^{2} \theta=1\) or \(\operatorname{cosec}^{2} \theta=1+\cot ^{2} \th...
Post a Comment
Read more