📝Problem:

Find the area between the curve and the $x$-axis between the given limits.

Given:

1. General Formula:

• The area under a curve for a parametric equation is given by:

For the given curve, the area is:

2. Substituting:

• Since $y$ is given in terms of t, we substitute:

Now we need to replace $dx$ with an expression in terms of $t$.

3. Finding $\frac{dx}{dt}$:

• Given $x = 1 + t$:

Therefore, $dx = dt$.

4. Adjusting the Integral:

• Replace $dx$ with dt in the integral:

5. Finding the Limits:

• The limits were initially for $dx$. We now convert them to limits for $t$:
• Lower limit: When $x = 2$, $2 = 1 + t$ gives $t = 1$.
• Upper limit: When $x = 6$, $6 = 1 + t$ gives $t = 5$.
• Therefore, the integral becomes:

6. Evaluating the Integral:

• Solve the integral:

• Simplify the expression:

📑Example: Find the Shaded Area

Given: