To find the area under a parametric curve, you can use the formula A = ∫ y dx, where x is the independent variable and y is the dependent variable. In this case, x=t^3-5t and y=7t^2. So, the area would be A = ∫ 7t^2 (t^3-5t) dt.
Calculating the area under a parametric curve can help us understand the behavior of the curve and make predictions about its future values. It can also be used to solve real-world problems, such as finding the distance traveled by an object with changing velocity over time.
To find the area under a parametric curve, you first need to find the limits of integration. This can be done by setting the two equations equal to each other and solving for t. Then, you can plug these limits into the formula A = ∫ y dx and integrate to find the area.
Negative values in the area calculation can arise when the curve dips below the x-axis. In this case, you would need to split the integral into two parts, one for the positive values and one for the negative values. Then, you can add the two areas together to get the total area.
Calculating the area under a parametric curve has many real-world applications. For example, it can be used to find the displacement of a moving object, the volume of a rotating object, or the work done by a changing force. It can also be used in fields such as engineering, physics, and economics.