Question

A curve has parametric equations x = at² , y =
2at. Find the area bounded by the curve, the x-
axis, and the ordinates at t = 1 and t = 2

step plz

Answer 1

The curve is just a parabola:

x = y^2/4a

So, the area is just

A = ∫[1,2] y(t) dx(t)
dx = 2at dt
A = ∫[1,2] 2at * 2at dt
= ∫[1,2] 4a^2t^2 dt
= 4/3 a^2 t^3 [1,2]
= 4/3 a^2 (8-1)
= 28/3 a^2

To check, express y as a function of x:

A = ∫[a,4a] y dx
= ∫[a,4a] 2√(ax) dx
= 2√a (2/3 x^(3/2)) [a,4a]
= 4/3√a (8a√a - a√a)
= 4/3 √a * 7a√a
= 28/3 a^2

Answer 2

X=at ² and y =2 at are parametric equations of