Final answer:
The area of the region bounded by the parametric curve x = t^2 − 2t, y = t, and the y-axis is found to be 2 square units by calculating the area of the right triangle formed between the curve and the y-axis.
Step-by-step explanation:
The student is asking how to find the area bounded by a parametric curve described by x = t2 − 2t, y = t, and the y-axis. To solve this problem, we first need to understand the behavior of the parametric function by plotting the curve or analyzing the functions to find the points where the curve intersects the y-axis. Since the y-axis is described by x = 0, we substitute this into the given equation for x to find the corresponding values of t.
To find these t values, solve the equation 0 = t2 − 2t. Rearrange to t2 − 2t = 0 and factoring yields t(t − 2) = 0. This gives us two t values, t1 = 0 and t2 = 2, which are the points where the curve intersects the y-axis. Next, to find the area under the curve, we need to find the area of the right triangle whose vertices are the origin, the point corresponding to t1, and the point corresponding to t2. The base of this triangle is along the y-axis, and the height is the distance from the origin to the point on the curve at t = 2.
Using the y function, we substitute t = 2 and get y = 2. This means our triangle has a height of 2. Since the base is along the y-axis, it is also 2 units long. The area of the triangle is then calculated with the formula Area = ½ × base × height, which gives us Area = ½ × 2 × 2 = 2 square units.