Interchange the order of integration and evaluate the integral [0.1]∫ [x,1]∫𝑒^𝑥/𝑦 𝑑𝑦𝑑𝑥
∫[0,1]∫[x,1] e^x/y dy dx
= ∫[0,1] e^x lny [x,1] dx
= ∫[0,1] e^x lnx dx
= -1.3179
Note that this integral cannot be done using elementary functions.
To change the order, note that the region involved is a triangle in the x-y plane, so we can write it instead as
∫[0,1]∫[y,0] e^x/y dx dy
= ∫[0,1] e^x/y [y,0] dy
= ∫[0,1] (1-e^y)/y dy
= -1.3179
To interchange the order of integration, we need to rewrite the integral in a form that allows us to integrate with respect to 𝑥 first and then 𝑦.
The given integral is:
[0.1]∫ [x,1]∫𝑒^𝑥/𝑦 𝑑𝑦𝑑𝑥
To interchange the order of integration, we will reverse the integration limits and integrate with respect to 𝑦 first:
∫ [0.1]∫ [y,1]𝑒^𝑥/𝑦 𝑑𝑥𝑑𝑦
Now, we can evaluate the inner integral with respect to 𝑥:
∫ [0.1]𝑒^𝑥/𝑦 |[y,1] 𝑑𝑦
Before proceeding, let's integrate 𝑒^𝑥/𝑦 with respect to 𝑥:
∫ 𝑒^𝑥/𝑦 𝑑𝑥 = 𝑦𝑒^𝑥/𝑦 + 𝐶
Now, substituting this result back into the equation:
∫ [0.1]𝑦𝑒^𝑥/𝑦 |[y,1] 𝑑𝑦
= ∫ [0.1](𝑦𝑒^𝑥/𝑦 + 𝐶) |[y,1] 𝑑𝑦
= ∫ [0.1](𝑒^𝑥 + 𝐶) |[y,1] 𝑑𝑦
Now, evaluate the integral with respect to 𝑦:
(𝑒^𝑥 + 𝐶)𝑦 |[y,1]
= (𝑒^𝑥 + 𝐶)(1) - (𝑒^𝑥 + 𝐶)(𝑦)
= (𝑒^𝑥 + 𝐶) - (𝑦𝑒^𝑥 + 𝑦𝐶)
Simplifying this expression will give you the final answer after interchanging the order of integration and evaluating the integral.