Final answer:
To find the area enclosed by the curve x(t) = t^2 - 3t, y(t) = t, and the y-axis, we need to compute the definite integral of y(t) with respect to x(t) over the given interval. We can find the upper limit of integration by solving the equation x(t) = 0 for t. Finally, we can calculate the integral to find the area enclosed by the curve and the y-axis.
Step-by-step explanation:
To find the area enclosed by the curve x(t) = t2 - 3t, y(t) = t, and the y-axis, we need to compute the definite integral of y(t) with respect to x(t) over the given interval. Since the curve intersects the y-axis at x = 0, we can find the upper limit of integration by solving the equation x(t) = 0 for t. Substituting x(t) = 0 into y(t), we get y(0) = 0. So, the definite integral of y(t) with respect to x(t) from x = 0 to x = t2 - 3t is the area enclosed by the curve and the y-axis.
Using the power rule, we can find x'(t) = 2t - 3 and y'(t) = 1. To calculate the integral, we need to express y(t) as a function of x(t). Solving the equation x(t) = t2 - 3t for t gives us t = 0 and t = 3. Therefore, the integral of y(t) with respect to x(t) from x = 0 to x = t2 - 3t is:
∫[0, t2 - 3t] y(t) dx(t) = ∫[0, t2 - 3t] y(t) (2t - 3) dt
Simplifying this integral will give us the area enclosed by the curve and the y-axis.