1.The curve of x=49-t^2 and y=t^3-1*t makes a loop which lies along the x-axis. What is the total area inside the loop?
2.Find the area of the region enclosed by the parametric equation: x=t^3-7*t and y=4*t^2.
1. To determine the total area inside the loop, we need to find the points where the curve intersects the x-axis.
The given equation x = 49 - t^2 represents a parabola that opens downward and is centered at (0,49). The curve intersects the x-axis when y = 0. So, we need to solve the equation t^3 - t = 0 for t.
Factoring out t, we have t(t^2 - 1) = 0.
Therefore, t = 0 or t^2 - 1 = 0.
Solving t^2 - 1 = 0, we get t = -1 or t = 1.
So, the curve intersects the x-axis at t = -1, 0, and 1.
To find the total area inside the loop, we need to calculate the definite integral of the curve's absolute value from t = -1 to t = 1.
Since the loop lies along the x-axis, the absolute value of y, |y|, represents the area between the curve and the x-axis.
Therefore, the total area inside the loop is given by the definite integral of |y| from t = -1 to t = 1.
The integral is ∫(from -1 to 1) |t^3 - t| dt.
Integrating the absolute value function requires breaking it down into two separate integrals:
∫(from -1 to 0) -(t^3 - t) dt + ∫(from 0 to 1) (t^3 - t) dt.
Evaluating the integrals, we find the total area inside the loop.
2. To find the area of the region enclosed by the parametric equations x = t^3 - 7t and y = 4t^2, we can use the formula for finding the area of a region bounded by a parametric curve.
The formula for finding the area enclosed by a parametric curve x = f(t) and y = g(t) over an interval a ≤ t ≤ b is:
A = ∫(from a to b)|x(t) * y'(t)| dt.
In this case, x(t) = t^3 - 7t and y(t) = 4t^2.
To find y'(t), we'll take the derivative of y(t) with respect to t:
y'(t) = 8t.
Now, substitute x(t) and y'(t) into the formula:
A = ∫(from a to b)|(t^3 - 7t) * (8t)| dt.
Simplify the integrand:
A = ∫(from a to b)|8t^4 - 56t^2| dt.
Evaluate the integral to find the area of the region enclosed by the parametric equations x and y.