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

8 In this question you must show detailed reasoning. A is the point \(( 1,0 ) , B\) is the point \(( 1,1 )\) and \(D\) is the point where the tangent to the curve \(y = x ^ { 3 }\) at B crosses the \(x\)-axis, as shown in Fig. 8. The tangent meets the \(y\)-axis at E. \begin{figure}[h] \end{figure}

1. Find the area of triangle ODE.
2. Find the area of the region bounded by the curve \(y = x ^ { 3 }\), the tangent at B and the \(y\)-axis.

9. \begin{figure}[h] \end{figure} Figure 5 shows a sketch of part of the curve \(C\) with equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = \frac { 6 x ^ { 2 } + 4 x - 2 } { 2 x + 1 } \quad x > - \frac { 1 } { 2 }$$

1. Find \(\mathrm { f } ^ { \prime } ( x )\), giving the answer in simplest form. The line \(l\) is the normal to \(C\) at the point \(P ( 2,6 )\)
2. Show that an equation for \(l\) is $$16 y + 5 x = 106$$
3. Write \(\mathrm { f } ( x )\) in the form \(A x + B + \frac { D } { 2 x + 1 }\) where \(A , B\) and \(D\) are constants. The region \(R\), shown shaded in Figure 5, is bounded by \(C , l\) and the \(x\)-axis.
4. Use algebraic integration to find the exact area of \(R\), giving your answer in the form \(P + Q \ln 3\), where \(P\) and \(Q\) are rational constants.
   (Solutions based entirely on calculator technology are not acceptable.)

7. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of part of the curve with equation $$y = \frac { 3 x - 1 } { x + 2 } \quad x > - 2$$

1. Show that $$\frac { 3 x - 1 } { x + 2 } \equiv A + \frac { B } { x + 2 }$$ where \(A\) and \(B\) are constants to be found. The finite region \(R\), shown shaded in Figure 4, is bounded by the curve, the line with equation \(x = 4\), the \(x\)-axis and the line with equation \(x = 1\) This region is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.
2. Use the answer to part (a) and algebraic integration to find the exact volume of the solid generated, giving your answer in the form $$\pi ( p + q \ln 2 )$$ where \(p\) and \(q\) are rational constants.

12 The diagram shows the curve \(y = \frac { x ^ { 2 } - 3 } { x + 1 }\) for \(x > - 1\). \includegraphics[max width=\textwidth, alt={}, center]{806dc286-416e-4785-8d13-0d524f808cb0-3_435_874_897_639}

1. Find the coordinates of the points where the curve crosses the axes.
2. Express \(\frac { x ^ { 2 } - 3 } { x + 1 }\) in the form \(A x + B + \frac { C } { x + 1 }\), where \(A , B\) and \(C\) are constants, and hence show that the exact area enclosed by the \(x\)-axis, the curve \(y = \frac { x ^ { 2 } - 3 } { x + 1 }\) and the lines \(x = 2\) and \(x = 4\) is \(4 + \ln \frac { 9 } { 25 }\).

12 A curve \(C\) is given by the parametric equations \(x = 2 \tan \theta , y = 1 + \operatorname { cosec } \theta\) for \(0 < \theta < 2 \pi , \theta \neq \frac { 1 } { 2 } \pi , \pi , \frac { 3 } { 2 } \pi\).

1. Show that the cartesian equation for \(C\) is \(\frac { 4 } { x ^ { 2 } } = y ^ { 2 } - 2 y\).
2. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence show that \(C\) has no stationary points.
3. \(P\) is the point on \(C\) where \(\theta = \frac { 1 } { 4 } \pi\). The tangent to \(C\) at \(P\) intersects the \(y\)-axis at \(Q\) and the \(x\)-axis at \(R\). Find the exact area of triangle \(O Q R\).

6 The diagram shows the curve with equation \(y = 7 x - 10 - x ^ { 2 }\) and the tangent to the curve at the point where \(x = 3\). \includegraphics[max width=\textwidth, alt={}, center]{69792771-6de6-4886-9c71-e794fcb7aaba-3_648_679_342_733}

1. Show that the curve crosses the \(x\)-axis at \(x = 2\).
2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence find the equation of the tangent to the curve at \(x = 3\). Show that the tangent crosses the \(x\)-axis at \(x = 1\).
3. Evaluate \(\int _ { 2 } ^ { 3 } \left( 7 x - 10 - x ^ { 2 } \right) \mathrm { d } x\) and hence find the exact area of the shaded region bounded by the curve, the tangent and the \(x\)-axis.

6 The diagram shows the curve with equation \(y = 7 x - 10 - x ^ { 2 }\) and the tangent to the curve at the point where \(x = 3\). \includegraphics[max width=\textwidth, alt={}, center]{29e924de-bedf-4719-bbfe-f5e0d3191d59-3_648_684_342_731}

1. Show that the curve crosses the \(x\)-axis at \(x = 2\).
2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence find the equation of the tangent to the curve at \(x = 3\). Show that the tangent crosses the \(x\)-axis at \(x = 1\).
3. Evaluate \(\int _ { 2 } ^ { 3 } \left( 7 x - 10 - x ^ { 2 } \right) \mathrm { d } x\) and hence find the exact area of the shaded region bounded by the curve, the tangent and the \(x\)-axis.

6 The diagram shows the curve with equation \(y = 7 x - 10 - x ^ { 2 }\) and the tangent to the curve at the point where \(x = 3\). \includegraphics[max width=\textwidth, alt={}, center]{0eb5bd24-e656-40f0-ad85-f21d3e2edf77-3_648_684_342_731}

1. Show that the curve crosses the \(x\)-axis at \(x = 2\).
2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence find the equation of the tangent to the curve at \(x = 3\). Show that the tangent crosses the \(x\)-axis at \(x = 1\).
3. Evaluate \(\int _ { 2 } ^ { 3 } \left( 7 x - 10 - x ^ { 2 } \right) \mathrm { d } x\) and hence find the exact area of the shaded region bounded by the curve, the tangent and the \(x\)-axis.

10

1. Show that \(\frac { \mathrm { d } } { \mathrm { d } x } ( \sin 3 x - 3 x \cos 3 x ) = 9 x \sin 3 x\). The curve shown in the figure below is part of the graph of the function \(y = x \sin 3 x\). \includegraphics[max width=\textwidth, alt={}, center]{3e4281d1-dbad-46a2-bbb7-97706bda2dfa-3_508_1136_1939_466}
2. Show that \(\int _ { 0 } ^ { \frac { 2 \pi } { 3 } } | x \sin 3 x | \mathrm { d } x = \frac { 4 \pi } { 9 }\).

2 Find the value of the positive constant \(k\) for which \(\int _ { 1 } ^ { k } ( 2 x - 1 ) \mathrm { d } x = 6\).

\includegraphics{figure_5} The diagram shows the curve with equation \(y = 10x^{\frac{1}{2}} - \frac{5}{2}x^{\frac{3}{2}}\) for \(x > 0\). The curve meets the \(x\)-axis at the points \((0, 0)\) and \((4, 0)\). Find the area of the shaded region. [4]

The curve with equation \(y = 2x - 8x^{\frac{1}{2}}\) has a minimum point at \(A\) and intersects the positive \(x\)-axis at \(B\). \begin{enumerate}[label=(\alph*)] \item Find the coordinates of \(A\) and \(B\). [4] \end enumerate}

\includegraphics{figure_6} The diagram shows the curve with equation \(y = 2x - 8x^{\frac{1}{2}}\) and the line \(AB\). It is given that the equation of \(AB\) is \(y = \frac{2x-32}{3}\). Find the area of the shaded region between the curve and the line. [5]

\includegraphics{figure_9} The diagram shows curves with equations \(y = 2x^{\frac{1}{2}} + 13x^{-\frac{1}{2}}\) and \(y = 3x^{-\frac{1}{4}} + 12\). The curves intersect at points \(A\) and \(B\).

1. Find the coordinates of \(A\) and \(B\). [4]
2. Hence find the area of the shaded region. [5]

\includegraphics{figure_9} The diagram shows part of the curve \(y = -x^2 + 8x - 10\) which passes through the points \(A\) and \(B\). The curve has a maximum point at \(A\) and the gradient of the line \(BA\) is \(2\).

1. Find the coordinates of \(A\) and \(B\). [7]
2. Find \(\int y \, dx\) and hence evaluate the area of the shaded region. [4]

\includegraphics{figure_2} The diagram shows the curve \(y = 2x^2\) and the points \(X(-2, 0)\) and \(P(p, 0)\). The point \(Q\) lies on the curve and \(PQ\) is parallel to the \(y\)-axis.

1. Express the area, \(A\), of triangle \(XPQ\) in terms of \(p\). [2]

The point \(P\) moves along the \(x\)-axis at a constant rate of 0.02 units per second and \(Q\) moves along the curve so that \(PQ\) remains parallel to the \(y\)-axis.

1. Find the rate at which \(A\) is increasing when \(p = 2\). [3]

\includegraphics{figure_10} The diagram shows part of the curve \(y = \frac{8}{\sqrt{(3x + 4)}}\). The curve intersects the \(y\)-axis at \(A(0, 4)\). The normal to the curve at \(A\) intersects the line \(x = 4\) at the point \(B\).

1. Find the coordinates of \(B\). [5]
2. Show, with all necessary working, that the areas of the regions marked \(P\) and \(Q\) are equal. [6]

\includegraphics{figure_7} The diagram shows two circles with centres \(A\) and \(B\) having radii 8 cm and 10 cm respectively. The two circles intersect at \(C\) and \(D\) where \(CAD\) is a straight line and \(AB\) is perpendicular to \(CD\).

1. Find angle \(ABC\) in radians. [1]
2. Find the area of the shaded region. [6]

\includegraphics{figure_10} The diagram shows part of the curve with equation \(y = (3x + 4)^{\frac{1}{3}}\) and the tangent to the curve at the point \(A\). The \(x\)-coordinate of \(A\) is 4.

1. Find the equation of the tangent to the curve at \(A\). [5]
2. Find, showing all necessary working, the area of the shaded region. [5]
3. A point is moving along the curve. At the point \(P\) the \(y\)-coordinate is increasing at half the rate at which the \(x\)-coordinate is increasing. Find the \(x\)-coordinate of \(P\). [3]

\includegraphics{figure_3} In the diagram, \(CXD\) is a semicircle of radius \(7\) cm with centre \(A\) and diameter \(CD\). The straight line \(YAX\) is perpendicular to \(CD\), and the arc \(CYD\) is part of a circle with centre \(B\) and radius \(8\) cm. Find the total area of the region enclosed by the two arcs. [6]

\includegraphics{figure_10} The diagram shows the line \(y = x + 1\) and the curve \(y = \sqrt{(x + 1)}\), meeting at \((-1, 0)\) and \((0, 1)\).

1. Find the area of the shaded region. [5]
2. Find the volume obtained when the shaded region is rotated through \(360°\) about the \(y\)-axis. [7]

\includegraphics{figure_9} The diagram shows parts of the graphs of \(y = x + 2\) and \(y = 3\sqrt{x}\) intersecting at points \(A\) and \(B\).

1. Write down an equation satisfied by the \(x\)-coordinates of \(A\) and \(B\). Solve this equation and hence find the coordinates of \(A\) and \(B\). [4]
2. Find by integration the area of the shaded region. [6]

\includegraphics{figure_11} The diagram shows part of the curve \(y = 3\sqrt{(4x + 1)} - 2x\). The curve crosses the \(y\)-axis at \(A\) and the stationary point on the curve is \(M\).

1. Obtain expressions for \(\frac{\text{d}y}{\text{d}x}\) and \(\int y \text{d}x\). [5]
2. Find the coordinates of \(M\). [3]
3. Find, showing all necessary working, the area of the shaded region. [4]

\includegraphics{figure_3} The diagram shows the curve with equation \(y = 8e^{-x} - e^{2x}\). The curve crosses the y-axis at the point A and the x-axis at the point B. The shaded region is bounded by the curve and the two axes.

1. Find the gradient of the curve at A. [3]
2. Show that the x-coordinate of B is \(\ln 2\) and hence find the area of the shaded region. [5]