Question

Please sketch the region of integration and evaluate the integral (see question).

Answer 1

\[\int\limits_{1}^{4}\int\limits_{0}^{\sqrt{x}}(3/2)e ^{y/\sqrt{x}}dydx\]

Answer 2

\[\huge \int\limits\limits_1^4 \left(\int\limits\limits_0^{\sqrt{x}} \frac{3}{2} e^{\frac{y}{\sqrt{x}}} \, dy\right) \, dx\]

Answer 3

well for inside integral, make substitution
u = y/sqrtx
du = dy/sqrtx
\[\rightarrow \frac{3}{2}\sqrt{x}\int\limits_{0}^{\sqrt{x}}e^{u} du = \frac{3}{2}\sqrt{x}(e-1)\]
then integrate that wrt x
\[ \frac{3}{2}(e-1)\int\limits_{0}^{4}\sqrt{x} dx = (e-1)*4^{3/2} = 8(e-1)\]

Answer 4

That looks right and is was I got too. Do you know how I should sketch the region of integration?

Answer 5

oops i made mistake on limits, i put 0 to 4 instead of 1 to 4

Answer 6

which changes answer to 7(e-1)

Answer 7

Oh right

Answer 8

this would be the region in xy plane