Show integral then evaluate area

5. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve with parametric equations $$x = \sqrt { 9 - 4 t } \quad y = \frac { t ^ { 3 } } { \sqrt { 9 + 4 t } } \quad 0 \leqslant t \leqslant \frac { 9 } { 4 }$$ The curve touches the \(x\)-axis when \(t = 0\) and meets the \(y\)-axis when \(t = \frac { 9 } { 4 }\) The region \(R\), shown shaded in Figure 2, is bounded by the curve, the \(x\)-axis and the \(y\)-axis.

1. Show that the area of \(R\) is given by $$K \int _ { 0 } ^ { \frac { 9 } { 4 } } \frac { t ^ { 3 } } { \sqrt { 81 - 16 t ^ { 2 } } } \mathrm {~d} t$$ where \(K\) is a constant to be found.
2. Using the substitution \(u = 81 - 16 t ^ { 2 }\), or otherwise, find the exact area of \(R\).
   (Solutions relying on calculator technology are not acceptable.)

8. \begin{figure}[h] \end{figure} The curve shown in Figure 2 has parametric equations $$x = t - 2 \sin t , \quad y = 1 - 2 \cos t , \quad 0 \leqslant t \leqslant 2 \pi$$

1. Show that the curve crosses the \(x\)-axis where \(t = \frac { \pi } { 3 }\) and \(t = \frac { 5 \pi } { 3 }\). The finite region \(R\) is enclosed by the curve and the \(x\)-axis, as shown shaded in Figure 2.
2. Show that the area of \(R\) is given by the integral $$\int _ { \frac { \pi } { 3 } } ^ { \frac { 5 \pi } { 3 } } ( 1 - 2 \cos t ) ^ { 2 } \mathrm {~d} t$$
3. Use this integral to find the exact value of the shaded area.

7. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve \(C\) with parametric equations $$x = 27 \sec ^ { 3 } t , y = 3 \tan t , \quad 0 \leqslant t \leqslant \frac { \pi } { 3 }$$

1. Find the gradient of the curve \(C\) at the point where \(t = \frac { \pi } { 6 }\)
2. Show that the cartesian equation of \(C\) may be written in the form $$y = \left( x ^ { \frac { 2 } { 3 } } - 9 \right) ^ { \frac { 1 } { 2 } } , \quad a \leqslant x \leqslant b$$ stating the values of \(a\) and \(b\).
   (3) \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{08f62966-2e63-4542-a10a-c6453a3215e7-10_581_1173_1628_475} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure} The finite region \(R\) which is bounded by the curve \(C\), the \(x\)-axis and the line \(x = 125\) is shown shaded in Figure 3. This region is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.
3. Use calculus to find the exact value of the volume of the solid of revolution. \section*{Question 7 continued} \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) 8. In an experiment testing solid rocket fuel, some fuel is burned and the waste products are collected. Throughout the experiment the sum of the masses of the unburned fuel and waste products remains constant. Let \(x\) be the mass of waste products, in kg , at time \(t\) minutes after the start of the experiment. It is known that at time \(t\) minutes, the rate of increase of the mass of waste products, in kg per minute, is \(k\) times the mass of unburned fuel remaining, where \(k\) is a positive constant. The differential equation connecting \(x\) and \(t\) may be written in the form $$\frac { \mathrm { d } x } { \mathrm {~d} t } = k ( M - x ) , \text { where } M \text { is a constant. }$$
   1. Explain, in the context of the problem, what \(\frac { \mathrm { d } x } { \mathrm {~d} t }\) and \(M\) represent. Given that initially the mass of waste products is zero,
   2. solve the differential equation, expressing \(x\) in terms of \(k , M\) and \(t\). Given also that \(x = \frac { 1 } { 2 } M\) when \(t = \ln 4\),
   3. find the value of \(x\) when \(t = \ln 9\), expressing \(x\) in terms of \(M\), in its simplest form. \section*{Question 8 continued}

8. \begin{figure}[h] \end{figure} Diagram not drawn to scale Figure 4 shows a sketch of part of the curve \(C\) with parametric equations $$x = 3 \theta \sin \theta , \quad y = \sec ^ { 3 } \theta , \quad 0 \leqslant \theta < \frac { \pi } { 2 }$$ The point \(P ( k , 8 )\) lies on \(C\), where \(k\) is a constant.

1. Find the exact value of \(k\). The finite region \(R\), shown shaded in Figure 4, is bounded by the curve \(C\), the \(y\)-axis, the \(x\)-axis and the line with equation \(x = k\).
2. Show that the area of \(R\) can be expressed in the form $$\lambda \int _ { \alpha } ^ { \beta } \left( \theta \sec ^ { 2 } \theta + \tan \theta \sec ^ { 2 } \theta \right) \mathrm { d } \theta$$ where \(\lambda , \alpha\) and \(\beta\) are constants to be determined.
3. Hence use integration to find the exact value of the area of \(R\).

6. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of the curve \(C\) with parametric equations $$x = 2 \cos 2 t \quad y = 4 \sin t \quad 0 \leqslant t \leqslant \frac { \pi } { 2 }$$ The region \(R\), shown shaded in Figure 3, is bounded by the curve, the \(x\)-axis and the \(y\)-axis.

1. 1. Show, making your working clear, that the area of \(R = \int _ { 0 } ^ { \frac { \pi } { 4 } } 32 \sin ^ { 2 } t \cos t d t\)
   2. Hence find, by algebraic integration, the exact value of the area of \(R\).
2. Show that all points on \(C\) satisfy \(y = \sqrt { a x + b }\), where \(a\) and \(b\) are constants to be found. The curve \(C\) has equation \(y = \mathrm { f } ( x )\) where f is the function $$f ( x ) = \sqrt { a x + b } \quad - 2 \leqslant x \leqslant 2$$ and \(a\) and \(b\) are the constants found in part (b).
3. State the range of f.

16. \begin{figure}[h] \end{figure} Figure 8 shows a sketch of the curve with parametric equations $$x = 4 \cos t \quad y = 2 \sin 2 t \quad 0 \leq t \leq \frac { \pi } { 2 }$$ where \(t\) is a parameter.
The finite region \(R\) is enclosed by the curve \(C\), the \(x\)-axis and the line \(x = 2\), as shown in Figure 7.
a. Show that the area of \(R\) is given by $$\int _ { \frac { \pi } { 3 } } ^ { \frac { \pi } { 2 } } 16 \sin ^ { 2 } t \cos t \mathrm {~d} t$$ b. Hence, using algebraic integration, find the exact area of \(R\), giving in the form \(a + b \sqrt { 3 }\), where \(a\) and \(b\) are constants to be determined.

16. \begin{figure}[h] \end{figure} Figure 6 shows a sketch of the curve \(C\) with parametric equations $$x = 8 \sin ^ { 2 } t \quad y = 2 \sin 2 t + 3 \sin t \quad 0 \leqslant t \leqslant \frac { \pi } { 2 }$$ The region \(R\), shown shaded in Figure 6, is bounded by \(C\), the \(x\)-axis and the line with equation \(x = 4\)

1. Show that the area of \(R\) is given by $$\int _ { 0 } ^ { a } \left( 8 - 8 \cos 4 t + 48 \sin ^ { 2 } t \cos t \right) \mathrm { d } t$$ where \(a\) is a constant to be found.
2. Hence, using algebraic integration, find the exact area of \(R\).

7. \begin{figure}[h] \end{figure} Figure 2 shows the curve with parametric equations $$x = \cos 2 t , \quad y = \operatorname { cosec } t , \quad 0 < t < \frac { \pi } { 2 } .$$ The point \(P\) on the curve has \(x\)-coordinate \(\frac { 1 } { 2 }\).

1. Find the value of the parameter \(t\) at \(P\).
2. Show that the tangent to the curve at \(P\) has the equation $$y = 2 x + 1$$ The shaded region is bounded by the curve, the coordinate axes and the line \(x = \frac { 1 } { 2 }\).
3. Show that the area of the shaded region is given by $$\int _ { \frac { \pi } { 6 } } ^ { \frac { \pi } { 4 } } k \cos t \mathrm {~d} t$$ where \(k\) is a positive integer to be found.
4. Hence find the exact area of the shaded region.
   7. continued
   7. continued

6 \includegraphics[max width=\textwidth, alt={}, center]{28beb431-45d5-4300-88fe-00d05d78790b-06_463_702_264_685} The diagram shows the curve \(C\) with parametric equations $$x = \frac { 1 } { 4 } \sin t , \quad y = t \cos t$$ where \(0 \leqslant t \leqslant k\).

1. Find the value of \(k\).
2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} t }\) in terms of \(t\). The maximum point on \(C\) is denoted by \(P\).
3. Using your answer to part (ii) and the standard small angle approximations, find an approximation for the \(x\)-coordinate of \(P\).
4. (a) Show that the area of the finite region bounded by \(C\) and the \(x\)-axis is given by $$b \int _ { 0 } ^ { a } t ( 1 + \cos 2 t ) \mathrm { d } t$$ where \(a\) and \(b\) are constants to be determined.
   (b) In this question you must show detailed reasoning. Hence find the exact area of the finite region bounded by \(C\) and the \(x\)-axis.

10. \begin{figure}[h] \end{figure} Figure 5 shows a sketch of the curve with parametric equations $$x = 3 t ^ { 2 } \quad y = \sin t \sin 2 t \quad 0 \leqslant t \leqslant \frac { \pi } { 2 }$$ The region \(R\), shown shaded in Figure 5, is bounded by the curve and the \(x\)-axis.

1. Show that the area of \(R\) is $$k \int _ { 0 } ^ { \frac { \pi } { 2 } } t \sin ^ { 2 } t \cos t \mathrm {~d} t$$ where \(k\) is a constant to be found.
2. Hence, using algebraic integration, find the exact area of \(R\), giving your answer in the form $$p \pi + q$$ where \(p\) and \(q\) are constants.

\includegraphics{figure_5} The diagram shows the curve \(C\) with parametric equations \(x = \frac{3}{t}\), \(y = t^2 e^{-2t}\), where \(t > 0\). The maximum point on \(C\) is denoted by \(P\).

1. Determine the exact coordinates of \(P\). [4] The shaded region \(R\) is enclosed by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 6\).
2. Show that the area of \(R\) is given by $$\int_a^b 3te^{-2t} dt,$$ where \(a\) and \(b\) are constants to be determined. [3]
3. Hence determine the exact area of \(R\). [5]

\includegraphics{figure_5} Figure 5 shows a sketch of the curve with parametric equations $$x = 3\cos 2t, \quad y = 2\tan t, \quad 0 \leq t \leq \frac{\pi}{4}.$$ The region \(R\), shown shaded in Figure 5, is bounded by the curve, the \(x\)-axis and the \(y\)-axis.

1. Show that the area of \(R\) is given $$\int_0^{\pi/4} 24\sin^2 t \, dt.$$ [4]
2. Hence, using algebraic integration, find the exact area of \(R\). [3]

\includegraphics{figure_6} The diagram shows the curve with parametric equations \(x = \ln(t^2 - 4)\), \(y = \frac{4}{t}\), where \(t > 2\). The shaded region \(R\) is enclosed by the curve, the \(x\)-axis and the lines \(x = \ln 5\) and \(x = \ln 12\).

1. Show that the area of \(R\) is given by $$\int_a^b \frac{8}{t(t^2 - 4)} dt,$$ where \(a\) and \(b\) are constants to be determined. [4]
2. In this question you must show detailed reasoning. Hence find the exact area of \(R\), giving your answer in the form \(\ln k\), where \(k\) is a constant to be determined. [8]
3. Find a cartesian equation of the curve in the form \(y = \text{f}(x)\). [3]