When dealing with parametric equations, figuring out the limits of integration is an important part of solving integrals.

Parametric equations describe a curve using a parameter, which we usually call $t$. For functions defined this way, we use two equations, $x(t)$ and $y(t)$, to show the coordinates of points along the curve.

How to Find Limits of Integration

1. Find the Range of the Parameter: The first thing you need to do is define the range of $t$. This range shows the interval over which you want to calculate the integral.

2. Identify Important Points: If you want to integrate over a certain part of the curve, you need to find the $t$ values that match those points. This usually means figuring out the $t$ values at the ends of the section of the curve you're looking at.

3. Check the Direction: It's also important to see if you're moving along the curve in a positive (forward) or negative (backward) direction as $t$ increases. This will affect how you set up your limits of integration and what you calculate in the end.

Example: Finding Limits of Integration

Let's say we have a parametric curve given by the equations:

$$x(t) = t^2 \quad \text{and} \quad y(t) = t^3$$

We want to find the area under the curve between $t = 0$ and $t = 2$.

1. Find the Range of the Parameter: Here, $t$ goes from $0$ to $2$.

2. Find the Corresponding Points: When $t = 0$, the point is $(0, 0)$. When $t = 2$, we find:

   $$x(2) = 2^2 = 4 \quad \text{and} \quad y(2) = 2^3 = 8$$

   So, the coordinates of this point are $(4, 8)$.

3. Determine the Direction: Since $t$ goes from $0$ to $2$, both $x(t)$ and $y(t)$ are increasing. This means we're moving from the starting point $(0, 0)$ towards the point $(4, 8)$.

Calculating the Integral

To find the area under the curve from $t=0$ to $t=2$, we can set up the integral as follows:

$$\text{Area} = \int_{0}^{2} y(t) \frac{dx}{dt} \, dt$$

Where $\frac{dx}{dt} = 2t$. So, the area becomes:

$$\text{Area} = \int_{0}^{2} (t^3)(2t) \, dt = \int_{0}^{2} 2t^4 \, dt$$

Now, we can solve this integral:

$$\text{Area} = 2 \left[ \frac{t^5}{5} \right]_{0}^{2} = 2 \left( \frac{32}{5} - 0 \right) = \frac{64}{5}$$

Conclusion

Finding the limits of integration for parametric equations involves a few clear steps: identifying the range of your parameter, locating the right points on the curve, and understanding the direction you're moving as the parameter changes. With careful evaluation, you can successfully calculate areas, arc lengths, and volumes related to curves defined by parametric equations.