1.08e Area between curve and x-axis: using definite integrals

1. The rectangular hyperbola \(H\) has equation \(x y = c ^ { 2 }\) where \(c\) is a positive constant.

The point \(P \left( c t , \frac { c } { t } \right)\), where \(t > 0\), lies on \(H\)

1. Use calculus to show that an equation of the normal to \(H\) at \(P\) is $$t ^ { 3 } x - t y = c \left( t ^ { 4 } - 1 \right)$$ The parabola \(C\) has equation \(y ^ { 2 } = 6 x\) The normal to \(H\) at the point with coordinates \(( 8,2 )\) meets \(C\) at the point \(Q\) where \(y > 0\)
2. Determine the exact coordinates of \(Q\) Given that
   • the point \(R\) is the focus of \(C\)
   • the line \(l\) is the directrix of \(C\)
   • the line through \(Q\) and \(R\) meets \(l\) at the point \(S\)
   • determine the exact length of \(Q S\)

6. The curve \(H\) has equation $$x y = a ^ { 2 } \quad x > 0$$ where \(a\) is a positive constant. The line with equation \(y = k x\), where \(k\) is a positive constant, intersects \(H\) at the point \(P\)

1. Use calculus to determine, in terms of \(a\) and \(k\), an equation for the tangent to \(H\) at \(P\) The tangent to \(H\) at \(P\) meets the \(x\)-axis at the point \(A\) and meets the \(y\)-axis at the point \(B\)
2. Determine the coordinates of \(A\) and the coordinates of \(B\), giving your answers in terms of \(a\) and \(k\)
3. Hence show that the area of triangle \(A O B\), where \(O\) is the origin, is independent of \(k\)

A parabola \(P\) has cartesian equation \(y ^ { 2 } = 28 x\). The point \(S\) is the focus of the parabola \(P\).

1. Write down the coordinates of the point \(S\). Points \(A\) and \(B\) lie on the parabola \(P\). The line \(A B\) is parallel to the directrix of \(P\) and cuts the \(x\)-axis at the midpoint of \(O S\), where \(O\) is the origin.
2. Find the exact area of triangle \(A B S\).
   

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.

5. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the curve with equation \(y = 4 x - x \mathrm { e } ^ { \frac { 1 } { 2 } x } , x \geqslant 0\) The curve meets the \(x\)-axis at the origin \(O\) and cuts the \(x\)-axis at the point \(A\) .

1. Find,in terms of \(\ln 2\) ,the \(x\) coordinate of the point \(A\) .
2. Find \(\int x \mathrm { e } ^ { \frac { 1 } { 2 } x } \mathrm {~d} x\) The finite region \(R\) ,shown shaded in Figure 2,is bounded by the \(x\)-axis and the curve with equation \(y = 4 x - x \mathrm { e } ^ { \frac { 1 } { 2 } x } , x \geqslant 0\)
3. Find,by integration,the exact value for the area of \(R\) . Give your answer in terms of \(\ln 2\) \includegraphics[max width=\textwidth, alt={}, center]{4de08317-5fb9-4789-8d57-ccf463224c78-18_2655_1943_114_118}

\begin{figure}[h] \end{figure} Figure 1 shows part of the curve \(C\) with equation \(y = ( 1 + x ) ( 4 - x )\).
The curve intersects the \(x\)-axis at \(x = - 1\) and \(x = 4\). The region \(R\), shown shaded in Figure 1, is bounded by \(C\) and the \(x\)-axis. Use calculus to find the exact area of \(R\).

\begin{figure}[h] \end{figure} Figure 1 shows part of the curve \(y = \frac { 3 } { \sqrt { } ( 1 + 4 x ) }\). The region \(R\) is bounded by the curve, the \(x\)-axis, and the lines \(x = 0\) and \(x = 2\), as shown shaded in Figure 1.

1. Use integration to find the area of \(R\). The region \(R\) is rotated \(360 ^ { \circ }\) about the \(x\)-axis.
2. Use integration to find the exact value of the volume of the solid formed.

1. The ellipse \(E\) has equation

$$\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 , \quad a > b > 0$$ The line \(l\) is a normal to \(E\) at a point \(P ( a \cos \theta , b \sin \theta ) , \quad 0 < \theta < \frac { \pi } { 2 }\)

1. Using calculus, show that an equation for \(l\) is $$a x \sin \theta - b y \cos \theta = \left( a ^ { 2 } - b ^ { 2 } \right) \sin \theta \cos \theta$$ The line \(l\) meets the \(x\)-axis at \(A\) and the \(y\)-axis at \(B\).
2. Show that the area of the triangle \(O A B\), where \(O\) is the origin, may be written as \(k \sin 2 \theta\), giving the value of the constant \(k\) in terms of \(a\) and \(b\).
3. Find, in terms of \(a\) and \(b\), the exact coordinates of the point \(P\), for which the area of the triangle \(O A B\) is a maximum.
   

5. The ellipse \(E\) has equation $$x ^ { 2 } + 9 y ^ { 2 } = 9$$ The point \(P ( a \cos \theta , b \sin \theta )\) is a general point on the ellipse \(E\).

1. Write down the value of \(a\) and the value of \(b\). The line \(L\) is a tangent to \(E\) at the point \(P\).
2. Show that an equation of the line \(L\) is given by $$3 y \sin \theta + x \cos \theta = 3$$ The line \(L\) meets the \(x\)-axis at the point \(Q\) and meets the \(y\)-axis at the point \(R\).
3. Show that the area of the triangle \(O Q R\), where \(O\) is the origin, is given by $$k \operatorname { cosec } 2 \theta$$ where \(k\) is a constant to be found. The point \(M\) is the midpoint of \(Q R\).
4. Find a cartesian equation of the locus of \(M\), giving your answer in the form \(y ^ { 2 } = \mathrm { f } ( x )\).

8 Fig. 8 shows part of the curve \(y = x \sin 3 x\). It crosses the \(x\)-axis at P . The point on the curve with \(x\)-coordinate \(\frac { 1 } { 6 } \pi\) is Q . \begin{figure}[h] \end{figure}

1. Find the \(x\)-coordinate of P .
2. Show that Q lies on the line \(y = x\).
3. Differentiate \(x \sin 3 x\). Hence prove that the line \(y = x\) touches the curve at Q .
4. Show that the area of the region bounded by the curve and the line \(y = x\) is \(\frac { 1 } { 72 } \left( \pi ^ { 2 } - 8 \right)\).

4 \includegraphics[max width=\textwidth, alt={}, center]{608720b6-5b18-45e9-8838-c94b347ab3b7-2_547_511_1813_817} The diagram shows a sketch of parts of the curves \(y = \frac { 16 } { x ^ { 2 } }\) and \(y = 17 - x ^ { 2 }\).

1. Verify that these curves intersect at the points \(( 1,16 )\) and \(( 4,1 )\).
2. Calculate the exact area of the shaded region between the curves.

7 \includegraphics[max width=\textwidth, alt={}, center]{608720b6-5b18-45e9-8838-c94b347ab3b7-3_563_639_1379_753} The diagram shows an equilateral triangle \(A B C\) with sides of length 12 cm . The mid-point of \(B C\) is \(O\), and a circular arc with centre \(O\) joins \(D\) and \(E\), the mid-points of \(A B\) and \(A C\).

1. Find the length of the arc \(D E\), and show that the area of the sector \(O D E\) is \(6 \pi \mathrm {~cm} ^ { 2 }\).
2. Find the exact area of the shaded region.

8 The cubic polynomial \(2 x ^ { 3 } + k x ^ { 2 } - x + 6\) is denoted by \(\mathrm { f } ( x )\). It is given that \(( x + 1 )\) is a factor of \(\mathrm { f } ( x )\).

1. Show that \(k = - 5\), and factorise \(\mathrm { f } ( x )\) completely.
2. Find \(\int _ { - 1 } ^ { 2 } f ( x ) \mathrm { d } x\).
3. Explain with the aid of a sketch why the answer to part (ii) does not give the area of the region between the curve \(y = \mathrm { f } ( x )\) and the \(x\)-axis for \(- 1 \leqslant x \leqslant 2\). \section*{[Question 9 is printed overleaf.]}

7 \includegraphics[max width=\textwidth, alt={}, center]{2ae05b46-6c9f-4aaa-9cba-1116c0ec27d4-3_579_557_858_794} The diagram shows part of the curve \(y = x ^ { 2 } - 3 x\) and the line \(x = 5\).

1. Explain why \(\int _ { 0 } ^ { 5 } \left( x ^ { 2 } - 3 x \right) \mathrm { d } x\) does not give the total area of the regions shaded in the diagram.
2. Use integration to find the exact total area of the shaded regions.

4 \includegraphics[max width=\textwidth, alt={}, center]{367db494-294e-4b53-b9e8-fd2a69fb6069-2_634_670_1123_740} The diagram shows the curve \(y = 4 - x ^ { 2 }\) and the line \(y = x + 2\).

1. Find the \(x\)-coordinates of the points of intersection of the curve and the line.
2. Use integration to find the area of the shaded region bounded by the line and the curve.

9

1. A tunnel is 100 m long. Its cross-section, shown in Fig. 9.1, is modelled by the curve $$y = \frac { 1 } { 4 } \left( 10 x - x ^ { 2 } \right) ,$$ where \(x\) and \(y\) are horizontal and vertical distances in metres. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{1e43ddbe-ae95-467b-a527-351ab8a4c4fe-004_506_812_676_653} \captionsetup{labelformat=empty} \caption{Figure 9.1}
   \end{figure} Using this model,
   (A) find the greatest height of the tunnel,
   (B) explain why \(100 \int _ { 0 } ^ { 10 } y \mathrm {~d} x\) gives the volume, in cubic metres, of earth removed to make the tunnel. Calculate this volume.
2. The roof of the tunnel is re-shaped to allow for larger vehicles. Fig. 9.2 shows the new crosssection. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{1e43ddbe-ae95-467b-a527-351ab8a4c4fe-004_513_1256_1894_575} \captionsetup{labelformat=empty} \caption{Fig. 9.2}
   \end{figure} Use the trapezium rule with 5 strips to estimate the new cross-sectional area.
   Hence estimate the volume of earth removed when the tunnel is re-shaped.

9

1. A tunnel is 100 m long. Its cross-section, shown in Fig. 9.1, is modelled by the curve $$y = \frac { 1 } { 4 } \left( 10 x - x ^ { 2 } \right) ,$$ where \(x\) and \(y\) are horizontal and vertical distances in metres. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{72b4624f-e716-4a37-96f3-01b46e0bd0fd-5_506_812_676_653} \captionsetup{labelformat=empty} \caption{Figure 9.1}
   \end{figure} Using this model,
   (A) find the greatest height of the tunnel,
   (B) explain why \(100 \int _ { 0 } ^ { 10 } y \mathrm {~d} x\) gives the volume, in cubic metres, of earth removed to make the tunnel. Calculate this volume.
2. The roof of the tunnel is re-shaped to allow for larger vehicles. Fig. 9.2 shows the new crosssection. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{72b4624f-e716-4a37-96f3-01b46e0bd0fd-5_513_1256_1894_575} \captionsetup{labelformat=empty} \caption{Fig. 9.2}
   \end{figure} Use the trapezium rule with 5 strips to estimate the new cross-sectional area.
   Hence estimate the volume of earth removed when the tunnel is re-shaped.

12 Fig. 12 is a sketch of the curve \(y = 2 x ^ { 2 } - 11 x + 12\). \begin{figure}[h] \end{figure}

1. Show that the curve intersects the \(x\)-axis at \(( 4,0 )\) and find the coordinates of the other point of intersection of the curve and the \(x\)-axis.
2. Find the equation of the normal to the curve at the point \(( 4,0 )\). Show also that the area of the triangle bounded by this normal and the axes is 1.6 units \(^ { 2 }\).
3. Find the area of the region bounded by the curve and the \(x\)-axis.

12

1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{15872003-2e41-47e9-a5bd-34e533768f8a-5_652_764_269_733} \captionsetup{labelformat=empty} \caption{Fig. 12}
   \end{figure} Fig. 12 shows part of the curve \(y = x ^ { 4 }\) and the line \(y = 8 x\), which intersect at the origin and the point P .
   (A) Find the coordinates of P , and show that the area of triangle OPQ is 16 square units.
   (B) Find the area of the region bounded by the line and the curve.
2. You are given that \(\mathrm { f } ( x ) = x ^ { 4 }\).
   (A) Complete this identity for \(\mathrm { f } ( x + h )\). $$f ( x + h ) = ( x + h ) ^ { 4 } = x ^ { 4 } + 4 x ^ { 3 } h + \ldots$$ (B) Simplify \(\frac { \mathrm { f } ( x + h ) - \mathrm { f } ( x ) } { h }\).
   (C) Find \(\lim _ { h \rightarrow 0 } \frac { \mathrm { f } ( x + h ) - \mathrm { f } ( x ) } { h }\).
   (D) State what this limit represents.

9 \begin{figure}[h] \end{figure} Fig. 9 shows a sketch of the graph of \(y = x ^ { 3 } - 10 x ^ { 2 } + 12 x + 72\).

1. Write down \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Find the equation of the tangent to the curve at the point on the curve where \(x = 2\).
3. Show that the curve crosses the \(x\)-axis at \(x = - 2\). Show also that the curve touches the \(x\)-axis at \(x = 6\).
4. Find the area of the finite region bounded by the curve and the \(x\)-axis, shown shaded in Fig. 9 . [4]

11 The cross-section of a brick wall built on horizontal ground is given, for \(0 \leq x \leq 6\), by the following function $$\begin{array} { l l } 0 \leq x \leq 2 & y = 1 \\ 2 \leq x \leq 4 & y = - \frac { 1 } { 2 } x ^ { 2 } + 3 x - 3 \\ 4 \leq x \leq 6 & y = 1 \end{array}$$ Units are metres.

1. Show that the highest point on the wall is 1.5 metres above the ground.
2. Find the area of the cross-section of the wall.

9 Fig. 9 shows \(P \quad\) The line \(y = x\) \(Q\) The curve \(y = \sqrt { \frac { 1 } { 2 } \left( x + x ^ { 2 } \right) }\) \(R \quad\) The curve \(\quad y = \sqrt { x }\). \begin{figure}[h] \end{figure}

1. Write down the area of the triangle formed by the line \(y = x\), the line \(x = 1\) and the \(x\)-axis.
2. Show that the area of the region formed by the curve \(y = \sqrt { x }\), the line \(x = 1\) and the \(x\)-axis is \(\frac { 2 } { 3 }\). An estimate is required of the Area, \(A\), of the region formed by the curve \(y = \sqrt { \frac { 1 } { 2 } \left( x + x ^ { 2 } \right) }\), the line \(x = 1\) and the \(x\)-axis.
3. Use results to parts (i) and (ii) to complete the statement $$\ldots \ldots \ldots \ldots . . < A < \ldots \ldots \ldots \ldots \ldots . .$$
4. Use the Trapezium Rule with 4 strips to find an estimate for \(A\).
5. Draw a sketch of Fig. 9. Use it to illustrate the area found as the trapezium rule estimate for \(A\).
   Explain how your diagram shows that the trapezium rule estimate must be:
   consistent with the answer to part (iv);
   an under-estimate for A .

1

1. Find \(\int \left( x ^ { 3 } - 2 x \right) \mathrm { d } x\). The graph below shows part of the curve \(y = x ^ { 3 } - 2 x\) for \(0 \leq x \leq 2\). \includegraphics[max width=\textwidth, alt={}, center]{c55a5f04-3573-4f36-a12c-3755bdd4a45b-2_528_1019_520_321}
2. Show that the area of the shaded region \(P\) is the same as the area of the shaded region \(Q\).

9. \includegraphics[max width=\textwidth, alt={}, center]{faa66f88-9bff-4dc9-955f-80cdab3fdd34-3_538_872_1790_447} The diagram shows the curves \(y = 2 x ^ { 2 } - 6 x - 3\) and \(y = 9 + 3 x - x ^ { 2 }\).

1. Find the coordinates of the points where the two curves intersect.
2. Find the area of the shaded region bounded by the two curves.

9. $$f ( x ) = x ^ { 3 } - 9 x ^ { 2 } + 24 x - 16$$

1. Evaluate \(\mathrm { f } ( 1 )\) and hence state a linear factor of \(\mathrm { f } ( x )\).
2. Show that \(\mathrm { f } ( x )\) can be expressed in the form $$\mathrm { f } ( x ) = ( x + p ) ( x + q ) ^ { 2 }$$ where \(p\) and \(q\) are integers to be found.
3. Sketch the curve \(y = \mathrm { f } ( x )\).
4. Using integration, find the area of the region enclosed by the curve \(y = \mathrm { f } ( x )\) and the \(x\)-axis.