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

7 The velocity of a particle moving in a straight line is modelled by \(\mathbf { v } = 0.6 \mathbf { t } ^ { 2 } - 2.1 \mathbf { t } + 1.5\) where \(v\) is the velocity in metres per second and \(t\) is the time in seconds.

1. Determine the times at which the particle is stationary.
2. Find the acceleration of the particle at the first of the times at which it is stationary.
3. Find the distance travelled by the particle between the times at which it is stationary.

6 In this question you must show detailed reasoning.
A particle moves in a straight line. Its velocity \(v \mathrm {~ms} ^ { - 1 }\) after \(t \mathrm {~s}\) is given by \(\mathrm { v } = \mathrm { t } ^ { 3 } - 5 \mathrm { t } ^ { 2 }\).

1. Find the times at which the particle is stationary.
2. Find the total distance travelled by the particle in the first 6 seconds.

10 In this question you must show detailed reasoning.
The equation of a curve is \(y = 12 x ^ { 3 } - 24 x ^ { 2 } - 60 x + 72\).
Determine the magnitude of the total area bounded by the curve and the \(x\)-axis.

6 Use integration to show that the area bounded by the \(x\)-axis and the curve with equation \(y = ( x - 1 ) ^ { 2 } ( x - 3 )\) is \(\frac { 4 } { 3 }\) square units.

3 Show that the area of the region bounded by the curve \(y = 3 x ^ { - \frac { 3 } { 2 } }\), the lines \(x = 1 , x = 3\) and the \(x\)-axis is \(6 - 2 \sqrt { 3 }\).

11 Fig. 11 shows the curve with parametric equations $$x = 2 \cos \theta , y = \sin \theta , 0 \leq \theta \leq 2 \pi .$$ The point P has parameter \(\frac { 1 } { 4 } \pi\). The tangent at P to the curve meets the axes at A and B . \begin{figure}[h] \end{figure}

1. Show that the equation of the line AB is \(x + 2 y = 2 \sqrt { 2 }\).
2. Determine the area of the triangle AOB .

7 A student is trying to find the binomial expansion of \(\sqrt { 1 - x ^ { 3 } }\).
She gets the first three terms as \(1 - \frac { x ^ { 3 } } { 2 } + \frac { x ^ { 6 } } { 8 }\).
She draws the graphs of the curves \(y = \sqrt { 1 - x ^ { 3 } } , y = 1 - \frac { x ^ { 3 } } { 2 }\) and \(y = 1 - \frac { x ^ { 3 } } { 2 } + \frac { x ^ { 6 } } { 8 }\) using software. \includegraphics[max width=\textwidth, alt={}, center]{c30a926b-d832-46f5-aa65-0066ef482c3d-6_901_1265_516_248}

1. Explain why \(1 - \frac { x ^ { 3 } } { 2 } + \frac { x ^ { 6 } } { 8 } \geqslant 1 - \frac { x ^ { 3 } } { 2 }\) for all values of \(x\).
2. Explain why the graphs suggest that the student has made a mistake in the binomial expansion.
3. Find the first four terms in the binomial expansion of \(\sqrt { 1 - x ^ { 3 } }\).
4. State the set of values of \(x\) for which the binomial expansion in part (c) is valid.
5. Sketch the curve \(y = 2.5 \sqrt { 1 - x ^ { 3 } }\) on the grid in the Printed Answer Booklet. \section*{(f) In this question you must show detailed reasoning.} The end of a bus shelter is modelled by the area between the curve \(\mathrm { y } = 2.5 \sqrt { 1 - x ^ { 3 } }\), the lines \(x = - 0.75 , x = 0.75\) and the \(x\)-axis. Lengths are in metres. Calculate, using your answer to part (c), an approximation for the area of the end of the bus shelter as given by this model.

9 The diagram shows the curve \(\mathrm { y } = 3 - \sqrt { \mathrm { x } }\). \includegraphics[max width=\textwidth, alt={}, center]{a0d9573f-8273-4562-a2d3-07f15d9da1af-6_810_1008_1155_283}

1. Draw the line \(\mathrm { y } = 5 \mathrm { x } - 1\) on the copy of the diagram in the Printed Answer Booklet.
2. In this question you must show detailed reasoning. Determine the exact area of the region bounded by the curve \(y = 3 - \sqrt { x }\), the lines \(y = 5 x - 1\) and \(x = 4\) and the \(x\)-axis.

8 The diagram shows the curve with equation \(y = 3 x ^ { 2 } - x ^ { 3 }\) and the line \(L\). \includegraphics[max width=\textwidth, alt={}, center]{b83c4e3a-36a6-4ca9-b44f-489676ca86d4-06_469_802_411_603} The points \(A\) and \(B\) have coordinates \(( - 1,0 )\) and \(( 2,0 )\) respectively. The curve touches the \(x\)-axis at the origin \(O\) and crosses the \(x\)-axis at the point \(( 3,0 )\). The line \(L\) cuts the curve at the point \(D\) where \(x = - 1\) and touches the curve at \(C\) where \(x = 2\).

1. Find the area of the rectangle \(A B C D\).
2. 1. Find \(\int \left( 3 x ^ { 2 } - x ^ { 3 } \right) \mathrm { d } x\).
   2. Hence find the area of the shaded region bounded by the curve and the line \(L\).
3. For the curve above with equation \(y = 3 x ^ { 2 } - x ^ { 3 }\) :
   1. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\);
   2. hence find an equation of the tangent at the point on the curve where \(x = 1\);
   3. show that \(y\) is decreasing when \(x ^ { 2 } - 2 x > 0\).
4. Solve the inequality \(x ^ { 2 } - 2 x > 0\).

4

1. The function f is defined for all values of \(x\) by \(\mathrm { f } ( x ) = x ^ { 3 } - 3 x ^ { 2 } - 6 x + 8\).
   1. Find the remainder when \(\mathrm { f } ( x )\) is divided by \(x + 1\).
   2. Given that \(\mathrm { f } ( 1 ) = 0\) and \(\mathrm { f } ( - 2 ) = 0\), write down two linear factors of \(\mathrm { f } ( x )\).
   3. Hence express \(x ^ { 3 } - 3 x ^ { 2 } - 6 x + 8\) as the product of three linear factors.
2. The curve with equation \(y = x ^ { 3 } - 3 x ^ { 2 } - 6 x + 8\) is sketched below. \includegraphics[max width=\textwidth, alt={}, center]{10bca9b4-5327-4b35-8b75-612b396e8a76-3_543_796_897_623}
   1. The curve intersects the \(y\)-axis at the point \(A\). Find the \(y\)-coordinate of \(A\).
   2. The curve crosses the \(x\)-axis when \(x = - 2\), when \(x = 1\) and also at the point \(B\). Use the results from part (a) to find the \(x\)-coordinate of \(B\).
3. 1. Find \(\int \left( x ^ { 3 } - 3 x ^ { 2 } - 6 x + 8 \right) d x\).
   2. Hence find the area of the shaded region bounded by the curve and the \(x\)-axis.

8 The diagram shows the curve with equation \(y = 3 x ^ { 2 } - x ^ { 3 }\) and the line \(L\). \includegraphics[max width=\textwidth, alt={}, center]{81f6fc30-982b-47b5-bab3-076cc0cc6563-5_479_816_406_596} The points \(A\) and \(B\) have coordinates \(( - 1,0 )\) and \(( 2,0 )\) respectively. The curve touches the \(x\)-axis at the origin \(O\) and crosses the \(x\)-axis at the point \(( 3,0 )\). The line \(L\) cuts the curve at the point \(D\) where \(x = - 1\) and touches the curve at \(C\) where \(x = 2\).

1. Find the area of the rectangle \(A B C D\).
2. 1. Find \(\int \left( 3 x ^ { 2 } - x ^ { 3 } \right) \mathrm { d } x\).
   2. Hence find the area of the shaded region bounded by the curve and the line \(L\).
3. For the curve above with equation \(y = 3 x ^ { 2 } - x ^ { 3 }\) :
   1. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\);
   2. hence find an equation of the tangent at the point on the curve where \(x = 1\);
   3. show that \(y\) is decreasing when \(x ^ { 2 } - 2 x > 0\).
4. Solve the inequality \(x ^ { 2 } - 2 x > 0\).

6

1. The polynomial \(\mathrm { p } ( x )\) is given by \(\mathrm { p } ( x ) = x ^ { 3 } + x - 10\).
   1. Use the Factor Theorem to show that \(x - 2\) is a factor of \(\mathrm { p } ( x )\).
   2. Express \(\mathrm { p } ( x )\) in the form \(( x - 2 ) \left( x ^ { 2 } + a x + b \right)\), where \(a\) and \(b\) are constants.
2. The curve \(C\) with equation \(y = x ^ { 3 } + x - 10\), sketched below, crosses the \(x\)-axis at the point \(Q ( 2,0 )\). \includegraphics[max width=\textwidth, alt={}, center]{22c93dd5-d96a-4e31-8507-9c802e386231-3_444_547_1781_756}
   1. Find the gradient of the curve \(C\) at the point \(Q\).
   2. Hence find an equation of the tangent to the curve \(C\) at the point \(Q\).
   3. Find \(\int \left( x ^ { 3 } + x - 10 \right) \mathrm { d } x\).
   4. Hence find the area of the shaded region bounded by the curve \(C\) and the coordinate axes.

6 The curve with equation \(y = 12 x ^ { 2 } - 19 x - 2 x ^ { 3 }\) is sketched below. \includegraphics[max width=\textwidth, alt={}, center]{2f7a8e95-4994-4732-a9a4-306c7b6cad92-3_444_819_1434_609} The curve crosses the \(x\)-axis at the origin \(O\), and the point \(A ( 2 , - 6 )\) lies on the curve.

1. 1. Find the gradient of the curve with equation \(y = 12 x ^ { 2 } - 19 x - 2 x ^ { 3 }\) at the point \(A\).
   2. Hence find the equation of the normal to the curve at the point \(A\), giving your answer in the form \(x + p y + q = 0\), where \(p\) and \(q\) are integers.
2. 1. Find the value of \(\int _ { 0 } ^ { 2 } \left( 12 x ^ { 2 } - 19 x - 2 x ^ { 3 } \right) \mathrm { d } x\).
   2. Hence determine the area of the shaded region bounded by the curve and the line \(O A\).

4 The curve sketched below passes through the point \(A ( - 2,0 )\). \includegraphics[max width=\textwidth, alt={}, center]{889639d6-0a31-4569-8370-1e72291a0c47-3_538_734_365_662} The curve has equation \(y = 14 - x - x ^ { 4 }\) and the point \(P ( 1,12 )\) lies on the curve.

1. 1. Find the gradient of the curve at the point \(P\).
   2. Hence find the equation of the tangent to the curve at the point \(P\), giving your answer in the form \(y = m x + c\).
2. 1. Find \(\int _ { - 2 } ^ { 1 } \left( 14 - x - x ^ { 4 } \right) \mathrm { d } x\).
   2. Hence find the area of the shaded region bounded by the curve \(y = 14 - x - x ^ { 4 }\) and the line \(A P\).
      (2 marks)

4 The curve with equation \(y = x ^ { 5 } - 3 x ^ { 2 } + x + 5\) is sketched below. The point \(O\) is at the origin and the curve passes through the points \(A ( - 1,0 )\) and \(B ( 1,4 )\). \includegraphics[max width=\textwidth, alt={}, center]{91170a77-e266-4c81-89ee-1fc29a538485-3_447_752_438_653}

1. Given that \(y = x ^ { 5 } - 3 x ^ { 2 } + x + 5\), find:
   1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\);
   2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
2. Find an equation of the tangent to the curve at the point \(A ( - 1,0 )\).
3. Verify that the point \(B\), where \(x = 1\), is a minimum point of the curve.
4. The curve with equation \(y = x ^ { 5 } - 3 x ^ { 2 } + x + 5\) is sketched below. The point \(O\) is at the origin and the curve passes through the points \(A ( - 1,0 )\) and \(B ( 1,4 )\). \includegraphics[max width=\textwidth, alt={}, center]{91170a77-e266-4c81-89ee-1fc29a538485-3_451_757_1736_648}
   1. Find \(\int _ { - 1 } ^ { 1 } \left( x ^ { 5 } - 3 x ^ { 2 } + x + 5 \right) \mathrm { d } x\).
   2. Hence find the area of the shaded region bounded by the curve between \(A\) and \(B\) and the line segments \(A O\) and \(O B\).

4 The curve with equation \(y = x ^ { 3 } - 5 x ^ { 2 } + 7 x - 3\) is sketched below. \includegraphics[max width=\textwidth, alt={}, center]{3729de55-7139-4f41-8584-640f173c0e09-3_444_588_411_717} The curve touches the \(x\)-axis at the point \(A ( 1,0 )\) and cuts the \(x\)-axis at the point \(B\).

1. 1. Use the factor theorem to show that \(x - 3\) is a factor of $$\mathrm { p } ( x ) = x ^ { 3 } - 5 x ^ { 2 } + 7 x - 3$$
   2. Hence find the coordinates of \(B\).
2. The point \(M\), shown on the diagram, is a minimum point of the curve with equation \(y = x ^ { 3 } - 5 x ^ { 2 } + 7 x - 3\).
   1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
   2. Hence determine the \(x\)-coordinate of \(M\).
3. Find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) when \(x = 1\).
4. 1. Find \(\int \left( x ^ { 3 } - 5 x ^ { 2 } + 7 x - 3 \right) \mathrm { d } x\).
   2. Hence determine the area of the shaded region bounded by the curve and the coordinate axes.

4 The curve with equation \(y = x ^ { 4 } - 8 x + 9\) is sketched below. \includegraphics[max width=\textwidth, alt={}, center]{66813123-3876-4484-aad1-4bfc09bb1508-5_410_609_383_721} The point \(( 2,9 )\) lies on the curve.

1. 1. Find \(\int _ { 0 } ^ { 2 } \left( x ^ { 4 } - 8 x + 9 \right) \mathrm { d } x\).
   2. Hence find the area of the shaded region bounded by the curve and the line \(y = 9\).
2. The point \(A ( 1,2 )\) lies on the curve with equation \(y = x ^ { 4 } - 8 x + 9\).
   1. Find the gradient of the curve at the point \(A\).
   2. Hence find an equation of the tangent to the curve at the point \(A\).

6 The diagram shows a curve and a line which intersect at the points \(A , B\) and \(C\). \includegraphics[max width=\textwidth, alt={}, center]{f2124c89-79de-4758-b7b8-ff273345b9dd-7_574_844_349_609} The curve has equation \(y = x ^ { 3 } - x ^ { 2 } - 5 x + 7\) and the straight line has equation \(y = x + 7\). The point \(B\) has coordinates ( 0,7 ).

1. 1. Show that the \(x\)-coordinates of the points \(A\) and \(C\) satisfy the equation $$x ^ { 2 } - x - 6 = 0$$
   2. Find the coordinates of the points \(A\) and \(C\).
2. Find \(\int \left( x ^ { 3 } - x ^ { 2 } - 5 x + 7 \right) \mathrm { d } x\).
3. Find the area of the shaded region \(R\) bounded by the curve and the line segment \(A B\).
   [0pt] [4 marks] \(7 \quad\) A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } - 10 x + 12 y + 41 = 0\). The point \(A ( 3 , - 2 )\) lies on the circle.

3 The diagram shows a sketch of a curve and a line. \includegraphics[max width=\textwidth, alt={}, center]{c7f38f7e-75aa-4b41-96fd-f38f968c225c-06_520_588_351_742} The curve has equation \(y = x ^ { 4 } + 3 x ^ { 2 } + 2\). The points \(A ( - 1,6 )\) and \(B ( 2,30 )\) lie on the curve.

1. Find an equation of the tangent to the curve at the point \(A\).
2. 1. Find \(\int _ { - 1 } ^ { 2 } \left( x ^ { 4 } + 3 x ^ { 2 } + 2 \right) \mathrm { d } x\).
   2. Calculate the area of the shaded region bounded by the curve and the line \(A B\).
      [0pt] [3 marks] \(4 \quad\) A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } + 2 x - 6 y - 40 = 0\).

7 The diagram shows the sketch of a curve and the tangent to the curve at \(P\). \includegraphics[max width=\textwidth, alt={}, center]{0d5b9235-af2b-4fd5-8fcf-b2b45e3c0a3c-14_519_817_356_614} The curve has equation \(y = 4 - x ^ { 2 } - 3 x ^ { 3 }\) and the point \(P ( - 2,24 )\) lies on the curve. The tangent at \(P\) crosses the \(x\)-axis at \(Q\).

1. 1. Find the equation of the tangent to the curve at the point \(P\), giving your answer in the form \(y = m x + c\).
   2. Hence find the \(x\)-coordinate of \(Q\).
2. 1. Find \(\int _ { - 2 } ^ { 1 } \left( 4 - x ^ { 2 } - 3 x ^ { 3 } \right) \mathrm { d } x\).
   2. The point \(R ( 1,0 )\) lies on the curve. Calculate the area of the shaded region bounded by the curve and the lines \(P Q\) and \(Q R\).
      [0pt] [3 marks]

8 A curve, drawn from the origin \(O\), crosses the \(x\)-axis at the point \(A ( 9,0 )\). Tangents to the curve at \(O\) and \(A\) meet at the point \(P\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{48c5470e-6489-4b25-98a6-1b4e101ab01c-006_763_879_466_577} The curve, defined for \(x \geqslant 0\), has equation $$y = x ^ { \frac { 3 } { 2 } } - 3 x$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. 1. Find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at the point \(O\) and hence write down an equation of the tangent at \(O\).
   2. Show that the equation of the tangent at \(A ( 9,0 )\) is \(2 y = 3 x - 27\).
   3. Hence find the coordinates of the point \(P\) where the two tangents meet.
3. Find \(\int \left( x ^ { \frac { 3 } { 2 } } - 3 x \right) \mathrm { d } x\).
4. Calculate the area of the shaded region bounded by the curve and the tangents \(O P\) and \(A P\).

8 A curve, drawn from the origin \(O\), crosses the \(x\)-axis at the point \(A ( 9,0 )\). Tangents to the curve at \(O\) and \(A\) meet at the point \(P\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{9fee4b6f-06e2-4ed8-8835-33ef33b98c94-5_778_901_461_571} The curve, defined for \(x \geqslant 0\), has equation $$y = x ^ { \frac { 3 } { 2 } } - 3 x$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. 1. Find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at the point \(O\) and hence write down an equation of the tangent at \(O\).
   2. Show that the equation of the tangent at \(A ( 9,0 )\) is \(2 y = 3 x - 27\).
   3. Hence find the coordinates of the point \(P\) where the two tangents meet.
3. Find \(\int \left( x ^ { \frac { 3 } { 2 } } - 3 x \right) \mathrm { d } x\).
4. Calculate the area of the shaded region bounded by the curve and the tangents \(O P\) and \(A P\).

5 A curve, drawn from the origin \(O\), crosses the \(x\)-axis at the point \(P ( 4,0 )\).
The normal to the curve at \(P\) meets the \(y\)-axis at the point \(Q\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{14c2acbb-5f3e-40e2-8b88-162341ab9f71-3_526_629_916_813} The curve, defined for \(x \geqslant 0\), has equation $$y = 4 x ^ { \frac { 1 } { 2 } } - x ^ { \frac { 3 } { 2 } }$$

1. 1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
      (3 marks)
   2. Show that the gradient of the curve at \(P ( 4,0 )\) is - 2 .
   3. Find an equation of the normal to the curve at \(P ( 4,0 )\).
   4. Find the \(y\)-coordinate of \(Q\) and hence find the area of triangle \(O P Q\).
   5. The curve has a maximum point \(M\). Find the \(x\)-coordinate of \(M\).
2. 1. Find \(\int \left( 4 x ^ { \frac { 1 } { 2 } } - x ^ { \frac { 3 } { 2 } } \right) \mathrm { d } x\).
   2. Find the total area of the region bounded by the curve and the lines \(P Q\) and \(Q O\).

4 The diagram shows a sketch of the curves with equations \(y = 2 x ^ { \frac { 3 } { 2 } }\) and \(y = 8 x ^ { \frac { 1 } { 2 } }\). \includegraphics[max width=\textwidth, alt={}, center]{0e19665b-5ee5-49e4-8de2-6c8dd17f61eb-3_433_720_1452_644} The curves intersect at the origin and at the point \(A\), where \(x = 4\).

1. 1. For the curve \(y = 2 x ^ { \frac { 3 } { 2 } }\), find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(x = 4\).
      (2 marks)
   2. Find an equation of the normal to the curve \(y = 2 x ^ { \frac { 3 } { 2 } }\) at the point \(A\).
2. 1. Find \(\int 8 x ^ { \frac { 1 } { 2 } } \mathrm {~d} x\).
   2. Find the area of the shaded region bounded by the two curves.
3. Describe a single geometrical transformation that maps the graph of \(y = 2 x ^ { \frac { 3 } { 2 } }\) onto the graph of \(y = 2 ( x + 3 ) ^ { \frac { 3 } { 2 } }\).
   (2 marks)

9 The diagram shows part of a curve crossing the \(x\)-axis at the origin \(O\) and at the point \(A ( 8,0 )\). Tangents to the curve at \(O\) and \(A\) meet at the point \(P\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{02e5dfac-18d7-480d-ac23-dfd2ca348cba-5_547_536_497_760} The curve has equation $$y = 12 x - 3 x ^ { \frac { 5 } { 3 } }$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. 1. Find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at the point \(O\) and hence write down an equation of the tangent at \(O\).
   2. Show that the equation of the tangent at \(A ( 8,0 )\) is \(y + 8 x = 64\).
3. Find \(\int \left( 12 x - 3 x ^ { \frac { 5 } { 3 } } \right) \mathrm { d } x\).
4. Calculate the area of the shaded region bounded by the curve from \(O\) to \(A\) and the tangents \(O P\) and \(A P\).