Question

$\blue{ \displaystyle \sf\int_{0}^{2} \int_{x }^{ {x}^{2} } x \cdot {y}^{2} - xdydx}$
​

Answer 1

$\displaystyle \sf\int_{0}^{2} \int_{x }^{ {x}^{2} } x \cdot {y}^{2} - xdydx$

Consider,

$\displaystyle \sf \int_{x }^{ {x}^{2} } x \cdot {y}^{2} - x \: dy$

$\implies \displaystyle \sf \int \: x {y}^{2} - x \: dy$

$\implies \displaystyle \sf \int \: x {y}^{2} \: dy \: - \int \: x \: dy$

$\sf \dfrac{x {y}^{3} }{3} - xy$

Return limits.

$\bigg( \sf\dfrac{x {y}^{3} }{x} - xy \bigg) \bigg|^{ {x}^{2} } _{x}$

$\displaystyle \sf \: \frac{x \times {( {x}^{2} )}^{3} }{3} - x \times {x}^{2} - \bigg( \frac{x \times {x}^{3} }{3} - x \times x \bigg)$

$\green{ \boxed{\displaystyle \sf \int_{x }^{ {x}^{2} } x \cdot {y}^{2} - x \: dy = \frac{ {x}^{7} - {x}^{4} }{3} - {x}^{3} + {x}^{2} - - - (1) }}$

$\displaystyle \sf\therefore \int_{0}^{ 2} \bigg( \frac{ {x}^{7} - {x}^{4} }{3} - {x}^{3} + {x}^{2} \bigg) \: dx \: - - - ( from \: (1))$

$\implies \displaystyle \sf \int \: \frac{ {x}^{7} - {x}^{4} }{3} - {x}^{3} + {x}^{2} \: dx$

$\displaystyle \sf \int \frac{ {x}^{7} - {x}^{4} }{3}dx - \int \: {x}^{3} \: dx + \int {x}^{2} dx$

$\implies \displaystyle \sf \frac{ {x}^{8} }{24} - \frac{ {x}^{5} }{15} - \frac{ {x}^{4} }{4} + \frac{ {x}^{3} }{3}$

$\displaystyle \sf \bigg( \frac{ {x}^{8} }{24} - \frac{ {x}^{5} }{15} - \frac{ {x}^{4} }{4} + \frac{ {x}^{3} }{3} \bigg) \bigg|^{2}_{0}$

$\implies \displaystyle \sf \frac{ {2}^{8} }{24} - \frac{ {2}^{5} }{15} - \frac{ {2}^{4} }{4} + \frac{ {2}^{3} }{3} - \bigg(0 \bigg)$

$\implies \displaystyle \sf \: \frac{256}{24} - \frac{32}{15} - 4 + \frac{8}{3}$

$\implies \displaystyle \sf \: \frac{36}{5}$

$\green{ \boxed{ \displaystyle \sf\int_{0}^{2} \int_{x }^{ {x}^{2} } x \cdot {y}^{2} - xdydx \: = \frac{36}{5} }}$

Additional Information -

$\begin{gathered}\boxed{\begin{array}{c|c}\bf f(x)&\bf\displaystyle\int\rm f(x)\:dx\\ \\ \frac{\qquad\qquad}{}&\frac{\qquad\qquad}{}\\ \rm k&\rm kx+C\\ \\ \rm sin(x)&\rm-cos(x)+C\\ \\ \rm cos(x)&\rm sin(x)+C\\ \\ \rm{sec}^{2}(x)&\rm tan(x)+C\\ \\ \rm{cosec}^{2}(x)&\rm-cot(x)+C\\ \\ \rm sec(x)\ tan(x)&\rm sec(x)+C\\ \\ \rm cosec(x)\ cot(x)&\rm-cosec(x)+C\\ \\ \rm tan(x)&\rm log(sec(x))+C\\ \\ \rm\dfrac{1}{x}&\rm log(x)+C\\ \\ \rm{e}^{x}&\rm{e}^{x}+C\\ \\ \rm x^{n},n\neq-1&\rm\dfrac{x^{n+1}}{n+1}+C\end{array}}\end{gathered}$

Some Properties -

$\displaystyle \rm\int \: f(x) \pm \: g(x) \: dx = \int \: f(x) \: dx \pm \int \: g(x) \: dx$

$\displaystyle \rm \int \: a \: dx = a \times x$

Answer 2

Step-by-step explanation:

$\displaystyle \sf\int_{0}^{2} \int_{x }^{ {x}^{2} } x \cdot {y}^{2} - xdydx$

Consider,

$\displaystyle \sf \int_{x }^{ {x}^{2} } x \cdot {y}^{2} - x \: dy$

$\implies \displaystyle \sf \int \: x {y}^{2} - x \: dy$

$\implies \displaystyle \sf \int \: x {y}^{2} \: dy \: - \int \: x \: dy$

$\sf \dfrac{x {y}^{3} }{3} - xy$

Return limits.

$\bigg( \sf\dfrac{x {y}^{3} }{x} - xy \bigg) \bigg|^{ {x}^{2} } _{x}$

$\displaystyle \sf \: \frac{x \times {( {x}^{2} )}^{3} }{3} - x \times {x}^{2} - \bigg( \frac{x \times {x}^{3} }{3} - x \times x \bigg)$

$\green{ \boxed{\displaystyle \sf \int_{x }^{ {x}^{2} } x \cdot {y}^{2} - x \: dy = \frac{ {x}^{7} - {x}^{4} }{3} - {x}^{3} + {x}^{2} - - - (1) }}$

$\displaystyle \sf\therefore \int_{0}^{ 2} \bigg( \frac{ {x}^{7} - {x}^{4} }{3} - {x}^{3} + {x}^{2} \bigg) \: dx \: - - - ( from \: (1))$

$\implies \displaystyle \sf \int \: \frac{ {x}^{7} - {x}^{4} }{3} - {x}^{3} + {x}^{2} \: dx$

$\displaystyle \sf \int \frac{ {x}^{7} - {x}^{4} }{3}dx - \int \: {x}^{3} \: dx + \int {x}^{2} dx$

$\implies \displaystyle \sf \frac{ {x}^{8} }{24} - \frac{ {x}^{5} }{15} - \frac{ {x}^{4} }{4} + \frac{ {x}^{3} }{3}$

$\displaystyle \sf \bigg( \frac{ {x}^{8} }{24} - \frac{ {x}^{5} }{15} - \frac{ {x}^{4} }{4} + \frac{ {x}^{3} }{3} \bigg) \bigg|^{2}_{0}$

$\implies \displaystyle \sf \frac{ {2}^{8} }{24} - \frac{ {2}^{5} }{15} - \frac{ {2}^{4} }{4} + \frac{ {2}^{3} }{3} - \bigg(0 \bigg)$

$\implies \displaystyle \sf \: \frac{256}{24} - \frac{32}{15} - 4 + \frac{8}{3}$

$\implies \displaystyle \sf \: \frac{36}{5}$

$\green{ \boxed{ \displaystyle \sf\int_{0}^{2} \int_{x }^{ {x}^{2} } x \cdot {y}^{2} - xdydx \: = \frac{36}{5} }}$

Additional Information -

$\begin{gathered}\boxed{\begin{array}{c|c}\bf f(x)&\bf\displaystyle\int\rm f(x)\:dx\\ \\ \frac{\qquad\qquad}{}&\frac{\qquad\qquad}{}\\ \rm k&\rm kx+C\\ \\ \rm sin(x)&\rm-cos(x)+C\\ \\ \rm cos(x)&\rm sin(x)+C\\ \\ \rm{sec}^{2}(x)&\rm tan(x)+C\\ \\ \rm{cosec}^{2}(x)&\rm-cot(x)+C\\ \\ \rm sec(x)\ tan(x)&\rm sec(x)+C\\ \\ \rm cosec(x)\ cot(x)&\rm-cosec(x)+C\\ \\ \rm tan(x)&\rm log(sec(x))+C\\ \\ \rm\dfrac{1}{x}&\rm log(x)+C\\ \\ \rm{e}^{x}&\rm{e}^{x}+C\\ \\ \rm x^{n},n\neq-1&\rm\dfrac{x^{n+1}}{n+1}+C\end{array}}\end{gathered}$

Some Properties -

$\displaystyle \rm\int \: f(x) \pm \: g(x) \: dx = \int \: f(x) \: dx \pm \int \: g(x) \: dx$

$\displaystyle \rm \int \: a \: dx = a \times x$