Question

Evaluate the double integral:

\(\int\limits_0^8(\int\limits_{y^{1/3}}^2\frac{y}{x^7+1}dx)dy\)

Answer 1

\(\int\limits_0^8(\int\limits_{y^{1/3}}^2\frac{y}{x^7+1}dx)dy\)

The region of integration is bounded between from x = y^1/3 to x = z

and from y = 0 to y = 8

⇒ y = x³

When integration limit change, it charged from y = 0 to y = x³

and x = 0 to x = 2

^{∴ \(\int\limits_0^8\int\limits_{y=1/3}^2\frac y{x^7+1}dxdy\) = \(\int\limits_0^2\int\limits_0^{x^3}\frac y{x^7+1}dy dx\)}

= \(\int\limits_0^2\frac1{x^7+1}(\frac{y^2}2)_0^{x^3}dx\) 

= \(\frac12\int\limits_0^2\frac{x^6}{x^7+1}dx\) 

= \(\frac1{14}[log(x^7+1)]_0^2\) 

 =  \(\frac1{14}(log(129 - log 1))\)

 = \(\frac1{14}\) log(129)