Questions: Find ∫ from 2 to 5 ∫ from 1 to 2 (x+y) dy dx

Solution Steps

To solve the given double integral, we need to integrate the inner integral with respect to \( y \) first, and then integrate the resulting expression with respect to \( x \).

We start with the double integral

\[ \int_{2}^{5} \int_{1}^{2} (x+y) \, dy \, dx. \]

First, we compute the inner integral

\[ \int_{1}^{2} (x+y) \, dy. \]

Calculating this gives:

\[ \int (x+y) \, dy = xy + \frac{y^2}{2} \bigg|_{1}^{2} = x(2) + \frac{2^2}{2} - \left( x(1) + \frac{1^2}{2} \right) = 2x + 2 - \left( x + \frac{1}{2} \right) = x + \frac{3}{2}. \]

Next, we compute the outer integral

\[ \int_{2}^{5} \left( x + \frac{3}{2} \right) \, dx. \]

Calculating this gives:

\[ \int \left( x + \frac{3}{2} \right) \, dx = \frac{x^2}{2} + \frac{3}{2}x \bigg|_{2}^{5} = \left( \frac{5^2}{2} + \frac{3}{2}(5) \right) - \left( \frac{2^2}{2} + \frac{3}{2}(2) \right). \]

Evaluating this results in:

\[ \left( \frac{25}{2} + \frac{15}{2} \right) - \left( 2 + 3 \right) = \frac{40}{2} - 5 = 20 - 5 = 15. \]

Final Answer