Question

Evaluate the double integral. $$ \int_{0}^{1} \int_{0}^{y}(x+y) d x d y $$

Short answer

The value of the double integral is 0.5.

Step-by-step solution

Integrate Inner Function With Respect to x

We begin by performing the integral with respect to x: \[\int_0^y (x + y) dx\] When integrating, we treat y as a constant. Thus, the integral becomes \[\frac{1}{2}x^2 + yx ]_0^y\] So, \(0.5*y^2 +y^2 - 0 = 1.5*y^2\)

Integrate the Result With Respect to y

We then perform the outer integral with respect to y, using the result above as the integrand: \[\int_0^1 1.5y^2 dy \] After integration, this becomes: \[0.5*y^3]_0^1\] So, \(0.5 - 0 = 0.5\)