Using line integrals to find area

Question

I am trying to find the area of a wall.
I have been given that the base is part of the circle of radius 1 centered at the origin, lying in the first quadrant.
I am also given that the height is equal to $$f(x, y) = y + 4x$$ How would I go about solving this?
I know that I have $$x^2 + y^2 = 1$$ and that I want dA using line integrals, but beyond this I'm a bit unsure as to what approach I should use.
Do I want to find $$\int \int y + 4x dx dy$$ with limits $$x = \pm \sqrt(1-y^2)$$ and $$y = \pm 1$$ or am I thinking about this in the wrong way? Thanks in advance

Answer 1

"The base is part of the circle of radius 1 centered at the origin, lying in the first quadrant" is parameterized as follow

$ \gamma :[0,\pi/2] \to \mathbb{R}^2 , \gamma(t) = (cos(t),sin(t))$

The height is equal to

$ f(x,y) = y+4x $

Then

$\int_{C} f \cdot \ d \gamma = \int_{0}^{\pi/2} f(\gamma(t))||\gamma'(t)|| \,dt$