Question

$\huge \underline{ \: \: \: \: \underline{\color {red}\bf QUESTION} \: \: \: \: }$ ​ ​

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

Answer 1

Given Question :-

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

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

Given curves are

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

and

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

Let assume that,

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

represents a upper parabola having vertex at (0, 5).

and

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

So, required area between the curves y = x² + 5 and y = x³ and the lines x = 1 and x = 2 is evaluated as

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

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

We know,

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

So, using this identity, we get

$\rm \:  =  \: \bigg[\dfrac{ {x}^{2 + 1} }{2 + 1} \bigg]_1^2 + \bigg[5x\bigg]_1^2 - \bigg[\dfrac{ {x}^{3 + 1} }{3 + 1} \bigg]_1^2$

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

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

$\rm \:  =  \: \bigg[\dfrac{8 - 1}{3} \bigg] + 5 - 4 + \bigg[\dfrac{1}{4} \bigg]$

$\rm \:  =  \: \bigg[\dfrac{7}{3} \bigg] + 1+ \bigg[\dfrac{1}{4} \bigg]$

$\rm \:  =  \: \dfrac{28 + 12 + 3}{12}$

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

Additional Information :-

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