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

\includegraphics{figure_4} The diagram shows the curves \(y = e^{3x}\) and \(y = (2x - 1)^4\). The shaded region is bounded by the two curves and the line \(x = \frac{1}{2}\). The shaded region is rotated completely about the \(x\)-axis. Find the exact volume of the solid produced. [7]

$$g(x) = 2x^3 + x^2 - 41x - 70$$

1. Use the factor theorem to show that \(g(x)\) is divisible by \((x - 5)\). [2]
2. Hence, showing all your working, write \(g(x)\) as a product of three linear factors. [4]

The finite region \(R\) is bounded by the curve with equation \(y = g(x)\) and the \(x\)-axis, and lies below the \(x\)-axis.

1. Find, using algebraic integration, the exact value of the area of \(R\). [4]

\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_4} Figure 4 shows a sketch of part of the curve with equation $$y = 2e^{2x} - xe^{2x}, \quad x \in \mathbb{R}$$ The finite region \(R\), shown shaded in Figure 4, is bounded by the curve, the \(x\)-axis and the \(y\)-axis. Use calculus to show that the exact area of \(R\) can be written in the form \(pe^t + q\), where \(p\) and \(q\) are rational constants to be found. (Solutions based entirely on graphical or numerical methods are not acceptable.) [5]

The diagram shows the curve \(y = 4 - x^2\) and the line \(y = x + 2\). \includegraphics{figure_3}

1. Find the \(x\)-coordinates of the points of intersection of the curve and the line. [2]
2. Use integration to find the area of the shaded region bounded by the line and the curve. [6]

The diagram shows the curve \(y = 4 - x^2\) and the line \(y = x + 2\). \includegraphics{figure_6}

1. Find the \(x\)-coordinates of the points of intersection of the curve and the line. [2]
2. Use integration to find the area of the shaded region bounded by the line and the curve. [6]

A region, R, is defined by \(x^2 - 8x + 12 \leq y \leq 12 - 2x\)

1. Sketch a graph to show the region R. Shade the region R.
2. Find the area of R [6 marks]

In this question you should show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. \includegraphics{figure_2} Figure 2 Figure 2 shows a sketch of part of the curve \(C\) with equation $$y = x^3 - 10x^2 + 27x - 23$$ The point \(P(5, -13)\) lies on \(C\) The line \(l\) is the tangent to \(C\) at \(P\)

1. Use differentiation to find the equation of \(l\), giving your answer in the form \(y = mx + c\) where \(m\) and \(c\) are integers to be found. [4]
2. Hence verify that \(l\) meets \(C\) again on the \(y\)-axis. [1]

The finite region \(R\), shown shaded in Figure 2, is bounded by the curve \(C\) and the line \(l\).

1. Use algebraic integration to find the exact area of \(R\). [4]

The diagram below represents the graph of the function \(y = (2x - 5)^4 - 1\) \includegraphics{figure_7}

1. Find the intersections of this graph with the \(x\) axis. [1]
2. Hence find the exact value of the area bounded by the curve and the \(x\) axis. [4]

\includegraphics{figure_7} Figure 7 shows the curves with equations $$y = kx^2 \quad x \geq 0$$ $$y = \sqrt{kx} \quad x \geq 0$$ where \(k\) is a positive constant. The finite region \(R\), shown shaded in Figure 7, is bounded by the two curves. Show that, for all values of \(k\), the area of \(R\) is \(\frac{1}{3}\) [5]

\includegraphics{figure_14} The diagram above shows the curve with equation $$y = (x-4)^2, \quad x \in \mathbb{R},$$ intersected by the straight line with equation \(y = 4\), at the points \(A\) and \(B\). The curve meets the \(y\) axis at the point \(C\). Calculate the exact area of the shaded region, bounded by the curve and the straight line segments \(AB\) and \(BC\). [8]

A curve \(C\) has equation \(y = f(x)\) where $$f(x) = x + 2\ln(e - x)$$

1. 1. Show that the equation of the normal to \(C\) at the point where \(C\) crosses the \(y\)-axis is given by $$y = \left(\frac{e}{2-e}\right)x + 2$$ [6 marks]
   2. Find the exact area enclosed by the normal and the coordinate axes. Fully justify your answer. [3 marks]

1. The equation \(f(x) = 0\) has one positive root, \(\alpha\).
   1. Show that \(\alpha\) lies between 2 and 3 Fully justify your answer. [3 marks]
   2. Show that the roots of \(f(x) = 0\) satisfy the equation $$x = e - e^{-\frac{x}{2}}$$ [2 marks]
   3. Use the recurrence relation $$x_{n+1} = e - e^{-\frac{x_n}{2}}$$ with \(x_1 = 2\) to find the values of \(x_2\) and \(x_3\) giving your answers to three decimal places. [2 marks]
   4. Figure 1 below shows a sketch of the graphs of \(y = e - e^{-\frac{x}{2}}\) and \(y = x\), and the position of \(x_1\) On Figure 1, draw a cobweb or staircase diagram to show how convergence takes place, indicating the positions of \(x_2\) and \(x_3\) on the \(x\)-axis. [2 marks] \includegraphics{figure_1}

A circle has centre \(C\) which lies on the \(x\)-axis, as shown in the diagram. The line \(y = x\) meets the circle at \(A\) and \(B\). The midpoint of \(AB\) is \(M\). \includegraphics{figure_9} The equation of the circle is \(x^2 - 6x + y^2 + a = 0\), where \(a\) is a constant.

1. In this question you must show detailed reasoning. Find the \(x\)-coordinate of \(M\) and hence show that the area of triangle \(ABC\) is \(\frac{3}{2}\sqrt{9 - 2a}\). [6]
2. 1. Find the value of \(a\) when the area of triangle \(ABC\) is zero. [1]
   2. Give a geometrical interpretation of the case in part (b)(i). [1]
3. Give a geometrical interpretation of the case where \(a = 5\). [1]

\includegraphics{figure_5} Figure 4 The line \(l_1\) has equation \(y = \frac{3}{5}x + 6\) The line \(l_2\) is perpendicular to \(l_1\) and passes through the point \(B(8, 0)\), as shown in the sketch in Figure 4.

1. Show that an equation for line \(l_2\) is $$5x + 3y = 40$$ [3]

Given that

• lines \(l_1\) and \(l_2\) intersect at the point C
• line \(l_1\) crosses the \(x\)-axis at the point A

1. find the exact area of triangle \(ABC\), giving your answer as a fully simplified fraction in the form \(\frac{p}{q}\) [5]

\includegraphics{figure_1} The diagram shows the curve \(y = 6x - x^2\) and the line \(y = 5\). Find the area of the shaded region. You must show detailed reasoning. [4]

The diagram shows the curve with equation \(y = 5x^4 + ax^3 + bx\), where \(a\) and \(b\) are integers. The curve has a minimum at the point \(P\) where \(x = 2\). \includegraphics{figure_1} The shaded region is enclosed by the curve, the \(x\)-axis and the line \(x = 2\). Given that the area of the shaded region is 48 units\(^2\), determine the \(y\)-coordinate of \(P\). [7]

\includegraphics{figure_8} The diagram shows the curve with equation \(y = 5x^4 + ax^3 + bx\), where \(a\) and \(b\) are integers. The curve has a minimum at the point \(P\) where \(x = 2\). The shaded region is enclosed by the curve, the \(x\)-axis and the line \(x = 2\). Given that the area of the shaded region is \(48\) units\(^2\), determine the \(y\)-coordinate of \(P\). [7]

A curve \(C\) has equation \(y = f(x)\) where $$f(x) = x + 2\ln(e - x)$$

1. 1. Show that the equation of the normal to \(C\) at the point where \(C\) crosses the \(y\)-axis is given by $$y = \left(\frac{e}{2-e}\right)x + 2$$ [6 marks]
   2. Find the exact area enclosed by the normal and the coordinate axes. Fully justify your answer. [3 marks]

1. The equation \(f(x) = 0\) has one positive root, \(\alpha\).
   1. Show that \(\alpha\) lies between 2 and 3 Fully justify your answer. [3 marks]
   2. Show that the roots of \(f(x) = 0\) satisfy the equation $$x = e - e^{\frac{x}{2}}$$ [2 marks]
   3. Use the recurrence relation $$x_{n+1} = e - e^{\frac{x_n}{2}}$$ with \(x_1 = 2\) to find the values of \(x_2\) and \(x_3\) giving your answers to three decimal places. [2 marks]
   4. Figure 1 below shows a sketch of the graphs of \(y = e - e^{\frac{x}{2}}\) and \(y = x\), and the position of \(x_1\) On Figure 1, draw a cobweb or staircase diagram to show how convergence takes place, indicating the positions of \(x_2\) and \(x_3\) on the \(x\)-axis. [2 marks] \includegraphics{figure_9}

In this question you must show detailed reasoning. The diagram shows the curve with equation \(y = \frac{x + 3}{\sqrt{x^2 + 9}}\). \includegraphics{figure_8} The region R, shown shaded in the diagram, is bounded by the curve, the \(x\)-axis, the \(y\)-axis, and the line \(x = 4\).

1. Determine the area of R. Give your answer in the form \(p + \ln q\) where \(p\) and \(q\) are integers to be determined. [6]

The region R is rotated through \(2\pi\) radians about the \(x\)-axis.

1. Determine the volume of the solid of revolution formed. Give your answer in the form \(\pi\left(a + b\ln\left(\frac{c}{d}\right)\right)\) where \(a\), \(b\), \(c\) and \(d\) are integers to be determined. [6]

\includegraphics{figure_1} The diagram shows the curve \(y = \sqrt{x - 3}\). The shaded region is bounded by the curve and the two axes. Find the exact area of the shaded region. [4]

The diagram shows the circle with centre O and radius 2, and the parabola \(y = \frac{1}{\sqrt{3}}(4 - x^2)\). \includegraphics{figure_5} The circle meets the parabola at points \(P\) and \(Q\), as shown in the diagram.

1. Verify that the coordinates of \(Q\) are \((1, \sqrt{3})\). [3]
2. Find the exact area of the shaded region enclosed by the arc \(PQ\) of the circle and the parabola. [8]

The curve \(y = x^3\) intersects the line \(y = kx\), \(k > 0\), at the origin and the point \(P\). The region bounded by the curve and the line, between the origin and \(P\), is denoted by \(R\).

1. Show that the area of the region \(R\) is \(\frac{1}{6}k^3\). [3]

The line \(x = a\) cuts the region \(R\) into two parts of equal area.

1. Show that \(k^3 - 6a^2k + 4a^3 = 0\). [3]

The gradient of the line \(y = kx\) increases at a constant rate with respect to time \(t\). Given that \(\frac{dk}{dt} = 2\),

1. determine the value of \(\frac{da}{dt}\) when \(a = 1\) and \(k = 2\), [4]
2. determine the value of \(\frac{da}{dt}\) when \(a = 1\) and \(k = 2\), expressing your answer in the form \(p + q\sqrt{3}\), where \(p\) and \(q\) are integers. [5]

A circle, of radius \(\sqrt{5}\) and centre the origin \(O\), is divided into two segments by the line \(y = 1\).

1. Determine the area of the smaller segment. [4]

The line is rotated clockwise about \(O\) through \(45^{\circ}\) and then reflected in the \(x\)-axis.

1. Find the equation of the line in its final position. [3]