A problem involving the same area was carried out by geometry without involving any integration. See the solution for the area of CED in the following link: Solution by plane geometry.
At this point, you can use your scientific calculator to solve for the area of region ABC. From calculator.
For the sake of discussion, integration is carried further step by step below.
When y = 0, θ = 0
When y = 10, θ = 30° = π/6