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

5 \includegraphics[max width=\textwidth, alt={}, center]{96a4df57-b3c7-4dbf-9bea-bb00ed6a4a16-2_512_775_1318_683} The diagram shows the curve \(y = \cos x\), for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\). A rectangle \(O A B C\) is drawn, where \(B\) is the point on the curve with \(x\)-coordinate \(\theta\), and \(A\) and \(C\) are on the axes, as shown. The shaded region \(R\) is bounded by the curve and by the lines \(x = \theta\) and \(y = 0\).

1. Find the area of \(R\) in terms of \(\theta\).
2. The area of the rectangle \(O A B C\) is equal to the area of \(R\). Show that $$\theta = \frac { 1 - \sin \theta } { \cos \theta }$$
3. Use the iterative formula \(\theta _ { n + 1 } = \frac { 1 - \sin \theta _ { n } } { \cos \theta _ { n } }\), with initial value \(\theta _ { 1 } = 0.5\), to determine the value of \(\theta\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

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1. Show that \(\cos ^ { 4 } \theta - \sin ^ { 4 } \theta \equiv \cos 2 \theta\).
2. Hence find the exact value of \(\int _ { - \frac { 1 } { 8 } \pi } ^ { \frac { 1 } { 8 } \pi } \left( \cos ^ { 4 } \theta - \sin ^ { 4 } \theta + 4 \sin ^ { 2 } \theta \cos ^ { 2 } \theta \right) \mathrm { d } \theta\).

9 \includegraphics[max width=\textwidth, alt={}, center]{98001cfe-46a1-4c8f-9230-c140ebff6176-14_535_1082_274_520} The diagram shows part of the curve \(y = ( 3 - x ) \mathrm { e } ^ { - \frac { 1 } { 3 } x }\) for \(x \geqslant 0\), and its minimum point \(M\).

1. Find the exact coordinates of \(M\).
2. Find the area of the shaded region bounded by the curve and the axes, giving your answer in terms of e.

8 \includegraphics[max width=\textwidth, alt={}, center]{3c63c42a-2658-4984-93e8-b2a8d18eb37a-12_473_839_274_644} The diagram shows part of the curve \(y = \sin \sqrt { x }\). This part of the curve intersects the \(x\)-axis at the point where \(x = a\).

1. State the exact value of \(a\).
2. Using the substitution \(u = \sqrt { x }\), find the exact area of the shaded region in the first quadrant bounded by this part of the curve and the \(x\)-axis.

6. \begin{figure}[h] \end{figure} \section*{In this question you must show all stages of your working.
Solutions relying on calculator technology are not acceptable.} Figure 3 shows

• the line \(l\) with equation \(y - 5 x = 75\)
• the curve \(C\) with equation \(y = 2 x ^ { 2 } + x - 21\)

The line \(l\) intersects the curve \(C\) at the points \(P\) and \(Q\), as shown in Figure 3 .

1. Find, using algebra, the coordinates of \(P\) and the coordinates of \(Q\). The region \(R\), shown shaded in Figure 3, is bounded by \(C , l\) and the \(x\)-axis.
2. Use inequalities to define the region \(R\).

5. Figure 2 Diagram NOT accurately drawn Figure 2 shows the plan view of a frame for a flat roof.
The shape of the frame consists of triangle \(A B D\) joined to triangle \(B C D\).
Given that

• \(B D = x \mathrm {~m}\)
• \(C D = ( 1 + x ) \mathrm { m }\)
• \(B C = 5 \mathrm {~m}\)
• angle \(B C D = \theta ^ { \circ }\)
  1. show that \(\cos \theta ^ { \circ } = \frac { 13 + x } { 5 + 5 x }\)

Given also that

14. \begin{figure}[h] \end{figure} Diagram NOT drawn to scale Figure 2 shows part of the line \(l\) with equation \(y = 2 x - 3\) and part of the curve \(C\) with equation \(y = x ^ { 2 } - 2 x - 15\) The line \(l\) and the curve \(C\) intersect at the points \(A\) and \(B\) as shown.

1. Use algebra to find the coordinates of \(A\) and the coordinates of \(B\). In Figure 2, the shaded region \(R\) is bounded by the line \(l\), the curve \(C\) and the positive \(x\)-axis.
2. Use integration to calculate an exact value for the area of \(R\).

15. \begin{figure}[h] \end{figure} Figure 5 shows a sketch of part of the curve \(C\) with equation $$y = x ^ { 3 } + 10 x ^ { \frac { 3 } { 2 } } + k x , \quad x \geqslant 0$$ where \(k\) is a constant.

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) The point \(P\) on the curve \(C\) is a minimum turning point.
   Given that the \(x\) coordinate of \(P\) is 4
2. show that \(k = - 78\) The line through \(P\) parallel to the \(x\)-axis cuts the \(y\)-axis at the point \(N\).
   The finite region \(R\), shown shaded in Figure 5, is bounded by \(C\), the \(y\)-axis and \(P N\).
3. Use integration to find the area of \(R\).

16. \begin{figure}[h] \end{figure} The straight line \(l\) with equation \(y = \frac { 1 } { 2 } x + 1\) cuts the curve \(C\), with equation \(y = x ^ { 2 } - 4 x + 3\), at the points \(P\) and \(Q\), as shown in Figure 2

1. Use algebra to find the coordinates of the points \(P\) and \(Q\). The curve \(C\) crosses the \(x\)-axis at the points \(T\) and \(S\).
2. Write down the coordinates of the points \(T\) and \(S\). The finite region \(R\) is shown shaded in Figure 2. This region \(R\) is bounded by the line segment \(P Q\), the line segment \(T S\), and the \(\operatorname { arcs } P T\) and \(S Q\) of the curve.
3. Use integration to find the exact area of the shaded region \(R\).

12. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of part of the curve \(C\) with equation $$y = \frac { 3 } { 4 } x ^ { 2 } - 4 \sqrt { x } + 7 , \quad x > 0$$ The point \(P\) lies on \(C\) and has coordinates \(( 4,11 )\).
Line \(l\) is the tangent to \(C\) at the point \(P\).

1. Use calculus to show that \(l\) has equation \(y = 5 x - 9\) The finite region \(R\), shown shaded in Figure 4, is bounded by the curve \(C\), the line \(x = 1\), the \(x\)-axis and the line \(l\).
2. Find, by using calculus, the area of \(R\), giving your answer to 2 decimal places.
   (Solutions based entirely on graphical or numerical methods are not acceptable.)

14. \begin{figure}[h] \end{figure} Figure 5 shows a sketch of part of the line \(l\) with equation \(y = 8 - x\) and part of the curve \(C\) with equation \(y = 14 + 3 x - 2 x ^ { 2 }\) The line \(l\) and the curve \(C\) intersect at the point \(A\) and the point \(B\) as shown.

1. Use algebra to find the coordinates of \(A\) and the coordinates of \(B\). The region \(R\), shown shaded in Figure 5, is bounded by the coordinate axes, the line \(l\), and the curve \(C\).
2. Use algebraic integration to calculate the exact area of \(R\).

14. \begin{figure}[h] \end{figure} The finite region \(R\), which is shown shaded in Figure 3, is bounded by the straight line \(l\) with equation \(y = 4 x + 3\) and the curve \(C\) with equation \(y = 2 x ^ { \frac { 3 } { 2 } } - 2 x + 3 , x \geqslant 0\) The line \(l\) meets the curve \(C\) at the point \(A\) on the \(y\)-axis and \(l\) meets \(C\) again at the point \(B\), as shown in Figure 3.

1. Use algebra to find the coordinates of \(A\) and \(B\).
2. Use integration to find the area of the shaded region \(R\).

16. \begin{figure}[h] \end{figure} Figure 6 shows a sketch of part of the curve \(C\) with equation $$y = x ( x - 1 ) ( x - 2 )$$ The point \(P\) lies on \(C\) and has \(x\) coordinate \(\frac { 1 } { 2 }\) The line \(l\), as shown on Figure 6, is the tangent to \(C\) at \(P\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
2. Use part (a) to find an equation for \(l\) in the form \(a x + b y = c\), where \(a\), \(b\) and \(c\) are integers. The finite region \(R\), shown shaded in Figure 6, is bounded by the line \(l\), the curve \(C\) and the \(x\)-axis. The line \(l\) meets the curve again at the point \(( 2,0 )\)
3. Use integration to find the exact area of the shaded region \(R\).

8. (a) Find \(\int \left( 3 x ^ { 2 } + 4 x - 15 \right) \mathrm { d } x\), simplifying each term. Given that \(b\) is a constant and $$\int _ { b } ^ { 4 } \left( 3 x ^ { 2 } + 4 x - 15 \right) \mathrm { d } x = 36$$ (b) show that \(b ^ { 3 } + 2 b ^ { 2 } - 15 b = 0\) (c) Hence find the possible values of \(b\).

12. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of part of the curve \(C\) with equation $$y = x ^ { 3 } - 9 x ^ { 2 } + 26 x - 18$$ The point \(P ( 4,6 )\) lies on \(C\).

1. Use calculus to show that the normal to \(C\) at the point \(P\) has equation $$2 y + x = 16$$ The region \(R\), shown shaded in Figure 4, is bounded by the curve \(C\), the \(x\)-axis and the normal to \(C\) at \(P\).
2. Show that \(C\) cuts the \(x\)-axis at \(( 1,0 )\)
3. Showing all your working, use calculus to find the exact area of \(R\).

15. \begin{figure}[h] \end{figure} A design for a logo consists of two finite regions \(R _ { 1 }\) and \(R _ { 2 }\), shown shaded in Figure 3 .
The region \(R _ { 1 }\) is bounded by the straight line \(l\) and the curve \(C\).
The region \(R _ { 2 }\) is bounded by the straight line \(l\), the curve \(C\) and the line with equation \(x = 5\) The line \(l\) has equation \(y = 8 x + 38\) The curve \(C\) has equation \(y = 4 x ^ { 2 } + 6\) Given that the line \(l\) meets the curve \(C\) at the points \(( - 2,22 )\) and \(( 4,70 )\), use integration to find

1. the area of the larger lower region, labelled \(R _ { 1 }\)
2. the exact value of the total area of the two shaded regions. Given that $$\frac { \text { Area of } R _ { 1 } } { \text { Area of } R _ { 2 } } = k$$
3. find the value of \(k\).

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16. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of the curve with equation \(y = 2 x ^ { 2 } - 11 x + 12\). The curve crosses the \(y\)-axis at the point \(A\) and crosses the \(x\)-axis at the points \(B\) and \(C\).

1. Find the coordinates of the points \(A , B\) and \(C\). The point \(D\) lies on the curve such that the line \(A D\) is parallel to the \(x\)-axis. The finite region \(R\), shown shaded in Figure 4, is bounded by the curve, the line \(A C\) and the line \(A D\).
2. Use algebraic integration to find the exact area of \(R\).

14. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of the curve \(C\) with equation \(y = - x ^ { 2 } + 6 x - 8\). The normal to \(C\) at the point \(P ( 5 , - 3 )\) is the line \(l\), which is also shown in Figure 3.

1. Find an equation for \(l\), giving your answer in the form \(a x + b y + c = 0\), where \(a\), b and \(c\) are integers. The finite region \(R\), shown shaded in Figure 3, is bounded below by the line \(l\) and the curve \(C\), and is bounded above by the \(x\)-axis.
2. Find the exact value of the area of \(R\).
   (Solutions based entirely on graphical or numerical methods are not acceptable.)

15. \begin{figure}[h] \end{figure} Figure 5 shows a sketch of part of the graph \(y = \mathrm { f } ( x )\), where $$f ( x ) = \frac { ( x - 3 ) ^ { 2 } ( x + 4 ) } { 2 } , \quad x \in \mathbb { R }$$ The graph cuts the \(y\)-axis at the point \(P\) and meets the positive \(x\)-axis at the point \(R\), as shown in Figure 5.

1. 1. State the \(y\) coordinate of \(P\).
   2. State the \(x\) coordinate of \(R\). The line segment \(P Q\) is parallel to the \(x\)-axis. Point \(Q\) lies on \(y = \mathrm { f } ( x ) , x > 0\)
2. Use algebra to show that the \(x\) coordinate of \(Q\) satisfies the equation $$x ^ { 2 } - 2 x - 15 = 0$$
3. Use part (b) to find the coordinates of \(Q\). The region \(S\), shown shaded in Figure 5, is bounded by the curve \(y = \mathrm { f } ( x )\) and the line segment \(P Q\).
4. Use calculus to find the exact area of \(S\).

10. \begin{figure}[h] \end{figure} The finite region \(R\), which is shown shaded in Figure 1, is bounded by the coordinate axes, the straight line \(l\) with equation \(y = \frac { 1 } { 3 } x + 5\) and the curve \(C\) with equation \(y = 4 x ^ { \frac { 1 } { 2 } } - x + 5 , x \geqslant 0\) The line \(l\) meets the curve \(C\) at the point \(D\) on the \(y\)-axis and at the point \(E\), as shown in Figure 1.

1. Use algebra to find the coordinates of the points \(D\) and \(E\). The curve \(C\) crosses the \(x\)-axis at the point \(F\).
2. Verify that the \(x\) coordinate of \(F\) is 25
3. Use algebraic integration to find the exact area of the shaded region \(R\).

11. \begin{figure}[h] \end{figure} The straight line with equation \(y = x + 4\) cuts the curve with equation \(y = - x ^ { 2 } + 2 x + 24\) at the points \(A\) and \(B\), as shown in Figure 2.

1. Use algebra to find the coordinates of the points \(A\) and \(B\). The finite region \(R\) is bounded by the straight line and the curve and is shown shaded in Figure 2.
2. Use calculus to find the exact area of \(R\).

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation $$y = 2 x ^ { 2 } + 7 \quad x \geqslant 0$$ The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the \(y\)-axis and the line with equation \(y = 17\) Find the exact area of \(R\).

9. \begin{figure}[h] \end{figure} Figure 2 shows

• the curve \(C\) with equation \(y = x - x ^ { 2 }\)
• the line \(l\) with equation \(y = m x\), where \(m\) is a constant and \(0 < m < 1\)

The line and the curve intersect at the origin \(O\) and at the point \(P\).

1. Find, in terms of \(m\), the coordinates of \(P\). The region \(R _ { 1 }\), shown shaded in Figure 2, is bounded by \(C\) and \(l\).
2. Show that the area of \(R _ { 1 }\) is $$\frac { ( 1 - m ) ^ { 3 } } { 6 }$$ The region \(R _ { 2 }\), also shown shaded in Figure 2, is bounded by \(C\), the \(x\)-axis and \(l\). Given that the area of \(R _ { 1 }\) is equal to the area of \(R _ { 2 }\)
3. find the exact value of \(m\). \includegraphics[max width=\textwidth, alt={}, center]{59c9f675-e7eb-47b9-b233-dfbe1844f792-33_108_76_2613_1875} \includegraphics[max width=\textwidth, alt={}, center]{59c9f675-e7eb-47b9-b233-dfbe1844f792-33_52_83_2722_1850}

1. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\) The table below shows some corresponding values of \(x\) and \(y\) for this curve.
The values of \(y\) are given to 3 decimal places. Using the trapezium rule with all the values of \(y\) in the given table,

1. obtain an estimate for $$\int _ { - 1 } ^ { 1 } \mathrm { f } ( x ) \mathrm { d } x$$ giving your answer to 2 decimal places.
2. Use your answer to part (a) to estimate
   1. \(\int _ { - 1 } ^ { 1 } ( \mathrm { f } ( x ) - 2 ) \mathrm { d } x\)
   2. \(\int _ { 1 } ^ { 3 } \mathrm { f } ( x - 2 ) \mathrm { d } x\)

1. In this question you must show all stages of your working.

\section*{Solutions based entirely on calculator technology are not acceptable.} \begin{figure}[h] \end{figure} Figure 3 shows

• the curve \(C\) with equation \(y = x ^ { 2 } - 4 x + 5\)
• the line \(l\) with equation \(y = 2\)

The curve \(C\) intersects the \(y\)-axis at the point \(D\).

1. Write down the coordinates of \(D\). The curve \(C\) intersects the line \(l\) at the points \(E\) and \(F\), as shown in Figure 3.
2. Find the \(x\) coordinate of \(E\) and the \(x\) coordinate of \(F\). Shown shaded in Figure 3 is
   • the region \(R _ { 1 }\) which is bounded by \(C , l\) and the \(y\)-axis
   • the region \(R _ { 2 }\) which is bounded by \(C\) and the line segments \(E F\) and \(D F\)
   Given that \(\frac { \text { area of } R _ { 1 } } { \text { area of } R _ { 2 } } = k\), where \(k\) is a constant,
3. use algebraic integration to find the exact value of \(k\), giving your answer as a simplified fraction.