Question

${{\boxed{\begin{array}{cc} \sf \: find \: the \: area \: of \: the \: region \: enclosed \\ \\ \sf \: by \: the \: following \: curves : \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \boxed{\bf \:y_1 = {e}^{x} } \\ \\ \boxed{\bf \: y_2 = {x}^{2} - 1} \\ \\ \boxed{ \bf \: x = 1} \\ \\ \boxed{ \bf \: x = - 1}\end{array}}}}$

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Answer 1

Answer:

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Answer 2

$\large\underline{\sf{Solution-}}$

$\rm :\longmapsto\:y = {e}^{x}$

represents an exponential curve passes through the point (0, 1).

and

$\rm :\longmapsto\:y = {x}^{2} - 1$

represents the upper parabola with vertex (0, - 1) and intersects the x - axis at ( - 1, 0 ) and ( 1, 0 ).

So,

Required area bounded between the curves

$\rm :\longmapsto\:y = {e}^{x}, \: y = {x}^{2} - 1, \: x = - 1 \: \: and \: \: x = 1 \: is$

$\rm \:  =  \:\displaystyle\int_{ - 1}^1 \: {e}^{x}dx \: + \: \displaystyle\int_{ - 1}^1 \: - ( {x}^{2} - 1) \: dx$

[ - ve sign because curve is below x - axis ]

$\rm \:  =  \:\displaystyle\int_{ - 1}^1 \: {e}^{x}dx \: + \: \displaystyle\int_{ - 1}^1 \: (-{x}^{2} + 1)\: dx$

$\rm \:  =  \: \bigg| {e}^{x} \bigg|_{ - 1}^1 + \bigg[x - \dfrac{ {x}^{3} }{3} \bigg]_{ - 1}^1$

$\rm \:  =  \:[e - {e}^{ - 1}] + [1 - ( - 1)] - \bigg[\dfrac{1}{3} - \dfrac{ - 1}{3} \bigg]$

$\rm \:  =  \:[e - {e}^{ - 1}] + [1 + 1] - \bigg[\dfrac{1}{3} + \dfrac{1}{3} \bigg]$

$\rm \:  =  \:[e - {e}^{ - 1}] + [2] - \bigg[\dfrac{2}{3} \bigg]$

$\rm \:  =  \:[e - {e}^{ - 1}] + 2 - \dfrac{2}{3}$

$\rm \:  =  \:[e - {e}^{ - 1}] + \dfrac{6 - 2}{3}$

$\rm \:  =  \:e - {e}^{ - 1}+ \dfrac{4}{3}$

$\rm \:  =  \:e - \dfrac{1}{e} + \dfrac{4}{3} \: square \: units$

More to know :-

$\begin{gathered}\begin{gathered}\boxed{\begin{array}{c|c} \bf f(x) & \bf \displaystyle \int \rm \:f(x) \: dx\\ \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf k & \sf kx + c \\ \\ \sf sinx & \sf - \: cosx+ c \\ \\ \sf cosx & \sf \: sinx + c\\ \\ \sf {sec}^{2} x & \sf tanx + c\\ \\ \sf {cosec}^{2}x & \sf - cotx+ c \\ \\ \sf secx \: tanx & \sf secx + c\\ \\ \sf cosecx \: cotx& \sf - \: cosecx + c\\ \\ \sf tanx & \sf logsecx + c\\ \\ \sf \dfrac{1}{x} & \sf logx+ c\\ \\ \sf {e}^{x} & \sf {e}^{x} + c\end{array}} \\ \end{gathered}\end{gathered}$