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

Matrices - Notes, Concept and All Important Formula

MATRICES 1. INTRODUCTION: A rectangular array of mn numbers in the form of \(m\) horizontal lines (called rows) and \(n\) vertical lines (called columns), is called a matrix of order \(m\) by \(n\) , written as \(m \times n\) matrix. In compact form, the matrix is represented by \(\mathrm{A}=\left[\mathrm{a}_{\mathrm{i}}\right]_{\mathrm{mx}}\) All Chapter Notes, Concept and Important Formula 2. SPECIAL TYPE OF MATRICES : (a) Row Matrix (Row vector) : \(A=\left[a_{11}, a_{12}, \ldots \ldots \ldots . a_{1 n}\right]\) i.e. row matrix has exactly one row. (b) Column Matrix (Column vector) : \(A=\left[\begin{array}{c}a_{11} \\ a_{21} \\ : \\ a_{m 1}\end{array}\right]\) i.e. columr matrix has exactly one column. (c) Zero or Null Matrix :  \(\left(A=O_{m \times n}\right)\) , An \(m \times n\) matrix whose all entries are zero. (d) Horizontal Matrix : A matrix of order \(\mathrm{m} \times \mathrm{n}\) is a horizontal matrix if \(\mathrm{n}>\mathrm{m}\) . (e) Vertical Matrix : ...
Post a Comment
Read more

Circle- Notes, Concept and All Important Formula

CIRCLE 1. DEFINITION : A circle is the locus of a point which moves in a plane in such a way that its distance from a fixed point remains constant. The fixed point is called the centre of the circle and the constant distance is called the radius of the circle. All Chapter Notes, Concept and Important Formula 2. STANDARD EQUATIONS OF THE CIRCLE : (a) Central Form: If \((\mathrm{h}, \mathrm{k})\) is the centre and \(\mathrm{r}\) is the radius of the circle then its equation is \((\mathbf{x}-\mathbf{h})^{2}+(\mathbf{y}-\mathbf{k})^{2}=\mathbf{r}^{2}\) (b) General equation of circle : \(\mathbf{x}^{2}+\mathbf{y}^{2}+\mathbf{2 g x}+\mathbf{2 f y}+\mathbf{c}=\mathbf{0}\) , where \(g, \mathrm{f}, c\) are constants and centre is \((-g,-f)\) i.e. \(\left(-\frac{\text { coefficient of } \mathrm{x}}{2},-\frac{\text { coefficient of } \mathrm{y}}{2}\right)\) and radius \(r=\sqrt{g^{2}+f^{2}-c}\) Note : The general quadratic equation in \(\mathrm{x}\) and \(\mathrm{y}\) , \(a x^{2}+b y^{2}+2 ...
Post a Comment
Read more

What is the integration of log (1+x^2)?

Use integration by part to solve this question. \[% \color{red}{\boxed{\color{blue}{\boxed{\color{black}{\text{ Using integration by parts, we have}\\\displaystyle \quad \int \ln \left(1+x^{2}\right) d x\\\displaystyle =x \ln \left(1+x^{2}\right)-\int x \frac{2 x}{1+x^{2}} d x \\\displaystyle =x \ln \left(1+x^{2}\right)-2 \int\left(1-\frac{1}{1+x^{2}}\right) d x \\\displaystyle =x \ln \left(1+x^{2}\right)-2 x+2 \tan ^{-1} x+C}}}}} \]
Post a Comment
Read more