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

AREA UNDER 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, 

$A=\displaystyle \int_{a}^{b} f(x) \,\, dx=\displaystyle \int_{a}^{b} y \,\, dx, f(x) \geq 0$

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All Chapter Notes, Concept and Important Formula

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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.

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

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4. Area between the curves $y=f(x)$ & $y=g(x)$ between the ordinates $x=a$ & $x=b$ is given by,

$A=\displaystyle \int_{a}^{b} f(x) \,\, dx-\displaystyle \int_{a}^{b} g(x) \,\, dx$

$=\displaystyle \int_{a}^{b}[f(x)-g(x)] \,\, dx, f(x) \geq g(x) \,\, \forall x \in(a, b)$

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

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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 $-\infty$.

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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.

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