Parametric loop enclosed area

12. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of part of the curve \(C\) with parametric equations $$x = 7 t ^ { 2 } - 5 , \quad y = t \left( 9 - t ^ { 2 } \right) , \quad t \in \mathbb { R }$$

1. Find an equation of the tangent to \(C\) at the point where \(t = 1\) Write your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers. The curve \(C\) cuts the \(x\)-axis at the points \(A\) and \(B\), as shown in Figure 3
2. 1. Find the \(x\) coordinate of the point \(A\).
   2. Find the \(x\) coordinate of the point \(B\). The region \(R\), shown shaded in Figure 3, is enclosed by the loop of the curve \(C\).
3. Use integration to find the area of \(R\).

7. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve \(C\) with parametric equations $$x = 5 t ^ { 2 } - 4 , \quad y = t \left( 9 - t ^ { 2 } \right)$$ The curve \(C\) cuts the \(x\)-axis at the points \(A\) and \(B\).

1. Find the \(x\)-coordinate at the point \(A\) and the \(x\)-coordinate at the point \(B\). The region \(R\), as shown shaded in Figure 2, is enclosed by the loop of the curve.
2. Use integration to find the area of \(R\).
   \section*{LU}

14 The curve \(C\) is defined for \(t \geq 0\) by the parametric equations $$x = t ^ { 2 } + t \quad \text { and } \quad y = 4 t ^ { 2 } - t ^ { 3 }$$ \(C\) is shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{042e248a-9efa-4844-957d-f05715900ffc-26_691_608_541_717} 14

1. Find the gradient of \(C\) at the point where it intersects the positive \(x\)-axis.
   14
2. (i) The area \(A\) enclosed between \(C\) and the \(x\)-axis is given by $$A = \int _ { 0 } ^ { b } y \mathrm {~d} x$$ Find the value of \(b\).
   14 (b) (ii) Use the substitution \(y = 4 t ^ { 2 } - t ^ { 3 }\) to show that $$A = \int _ { 0 } ^ { 4 } \left( 4 t ^ { 2 } + 7 t ^ { 3 } - 2 t ^ { 4 } \right) \mathrm { d } t$$ 14 (b) (iii) Find the value of \(A\).