# Thread: Double Integral over a region

1. ## Double Integral over a region

Question: Let $\displaystyle R$ be the region in the plane defined by:

$\displaystyle R = \{(x,y) \in R^2 : 1\leq x \leq 2, x\leq y \leq x^3 \}$

Calculate:

$\displaystyle \int \int_R e^{\frac{y}{x}} dA$

My attempt:

$\displaystyle \int_1^2 \int_x^x^3 e^{\frac{y}{x}} dy dx$ $\displaystyle =$ $\displaystyle \int_1^2 x e^x^2 - xe dx$

$\displaystyle =$ $\displaystyle \frac{1}{2}[e^x^2 - x^2e]_1^2$ $\displaystyle =$ $\displaystyle \frac{1}{2}e^4 - 2e$

2. ## Re: Double Integral over a region

Originally Posted by M.R
Question: Let $\displaystyle R$ be the region in the plane defined by:

$\displaystyle R = \{(x,y) \in R^2 : 1\leq x \leq 2, x\leq y \leq x^3 \}$

Calculate:

$\displaystyle \int \int_R e^{\frac{y}{x}} dA$

My attempt:

$\displaystyle \int_1^2 \int_x^x^3 e^{\frac{y}{x}} dy dx$ $\displaystyle =$ $\displaystyle \int_1^2 x e^x^2 - xe dx$

$\displaystyle =$ $\displaystyle \frac{1}{2}[e^x^2 - x^2e]_1^2$ $\displaystyle =$ $\displaystyle \frac{1}{2}e^4 - 2e$