Questions: After evaluating the inner integral, we have the integral ∫0^2 ∫0^y^3 e^x / y d x d y=∫0^2 g(y) d y where g(y)=

Transcript text: After evaluating the inner integral, we have the integral \[ \int_{0}^{2} \int_{0}^{y^{3}} e^{x / y} d x d y=\int_{0}^{2} g(y) d y \] where \[ g(y)= \]

Solution

Solution Steps

To find \( g(y) \), we need to evaluate the inner integral \(\int_{0}^{y^{3}} e^{x / y} d x\). This involves integrating the function \( e^{x/y} \) with respect to \( x \) from 0 to \( y^3 \). After finding this expression, it will represent \( g(y) \).

Step 1: Evaluate the Inner Integral

We need to evaluate the inner integral

\[ g(y) = \int_{0}^{y^{3}} e^{x/y} \, dx. \]

Using integration, we find that

\[ g(y) = y e^{y^2} - y. \]

Step 2: Simplify the Expression

The expression for \( g(y) \) can be factored as follows:

\[ g(y) = y(e^{y^2} - 1). \]