Question

Evaluate the iterated integral \(\int_0^1 {\int_{{x^2}}^x {(1 + 2y)} } dydx. \)

Short answer

The value of integral is \(\frac{3}{{10}}\).

Step-by-step solution

Integrating with respect to \(y\) and treating \(x\) as a constant.

\(\int_{{x^2}}^x {(1 + 2y)} dy = \left( {y + \frac{{2{y^2}}}{2}} \right)_{{x^2}}^x\)

\( = \left( {y + {y^2}} \right)_{{x^2}}^x\)

\( = \left( {x + {x^2}} \right) - \left( {{x^2} + {x^4}} \right)\)

\( = x - {x^4}\)

Now, integrating write respect to.

\(\int_0^1 {\int_{{x^2}}^x {(1 + 2y)} } dydx = \int_0^1 {\left( {x - {x^4}} \right)} dx\)

\( = \left( {\frac{{{x^2}}}{2} - \frac{{{x^5}}}{5}} \right)_0^1\)

\( = \frac{1}{2} - \frac{1}{5}\)

\( = \frac{3}{{10}}\)

Therefore, Value of integral is \(\frac{3}{{10}}\)