Question

Find the area between the curves y = x² + 5 and y = x³ and the lines x = 1 and x = 2​

Answer 1

consider the given curves

$\tt y = x^{3}$

$\tt y = x^{2}$

Area bounded the these curves

$\tt = |f^{x _2 } _ {x^{1}}( {y}^{2} - {y}^{1} dx |$

$\tt | f^{2} _ {1} (x^{2}-x{3}) do |$

$\tt | [ \frac{ x^{3} }{ x } ^{2} _{1} - ( \frac{ x^{4} }{ 4 }) ^{2} _{1} |$

$\tt | \frac{ 1 }{3 } (8-1) - \frac{ 1}{4} ( 16-1 ) |$

$\tt | \frac{ 7 }{3} = \frac{15}{4} |$

$\tt | \frac{ 28-45 }{12} |$

$\tt \frac{ 17 }{12} ^{2} units$

Hence, the area bounded by these curves is

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$\LARGE{\pmb{\frak {\frac{17 }{12} ^{2} units }}}$

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Answer 2

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

Given curves are

$\rm :\longmapsto\:y = {x}^{2} + 5$

and

$\rm :\longmapsto\:y = {x}^{3}$

with boundary conditions,

$\rm :\longmapsto\:x = 1 \: \: and \: \: x = 2$

Let assume that,

$\rm :\longmapsto\:y_1 = {x}^{2} + 5$

and

$\rm :\longmapsto\:y_2 = {x}^{3}$

So, required area between the curves is given by

$\rm \: = \:\displaystyle\int_1^2\rm (y_1 - y_2) \: dx$

$\rm \: = \:\displaystyle\int_1^2\rm ( {x}^{2} + 5 - {x}^{3} ) \: dx$

We know,

$\red{\boxed{ \rm{ \: \displaystyle\int \: {x}^{n} \: dx \: = \: \frac{ {x}^{n + 1} }{n + 1} + c \: \: }}}$

So, using this identity, we get

$\rm \: = \:\bigg[\dfrac{ {x}^{3} }{3} + 5x - \dfrac{ {x}^{4} }{4} \bigg]_1^2\rm$

$\rm \: = \:\dfrac{8 - 1}{3} + 5(2 - 1) - \dfrac{16 - 1}{4}$

$\rm \: = \:\dfrac{7}{3} + 5 - \dfrac{15}{4}$

$\rm \: = \:\dfrac{28 + 60 - 45}{12}$

$\rm \: = \:\dfrac{43}{12} \: square \: units$

Additional Information :-

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