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

\includegraphics{figure_8} The diagram shows the curve \(M\) with equation \(y = xe^{-2x}\).

1. Show that \(M\) has a point of inflection at the point \(P\) where \(x = 1\). [5]

The line \(L\) passes through the origin \(O\) and the point \(P\). The shaded region \(R\) is enclosed by the curve \(M\) and the line \(L\).

1. Show that the area of \(R\) is given by $$\frac{1}{4}(a + be^{-2}),$$ where \(a\) and \(b\) are integers to be determined. [6]

In this question you must show detailed reasoning. \includegraphics{figure_6} The diagram shows the curves \(y = \sqrt{2x + 9}\) and \(y = 4\mathrm{e}^{-2x} - 1\) which intersect on the \(y\)-axis. The shaded region is bounded by the curves and the \(x\)-axis. Determine the area of the shaded region, giving your answer in the form \(p + q \ln 2\) where \(p\) and \(q\) are constants to be determined. [8]

A curve has equation \(y = a^2 - x^2\), where \(a > 0\) The area enclosed between the curve and the \(x\)-axis is 36 units. Find the value of \(a\). Fully justify your answer. [6 marks]

A curve has equation $$y = x - a\sqrt{x} + b$$ where \(a\) and \(b\) are constants. The curve intersects the line \(y = 2\) at points with coordinates \((1, 2)\) and \((9, 2)\), as shown in the diagram below. \includegraphics{figure_1}

1. Show that \(a\) has the value 4 and find the value of \(b\) [3 marks]
2. On the diagram, the region enclosed between the curve and the line \(y = 2\) is shaded. Show that the area of this shaded region is \(\frac{16}{3}\) Fully justify your answer. [6 marks]

A curve has equation \(y = 6x^2 + \frac{8}{x^2}\) and is sketched below for \(x > 0\) \includegraphics{figure_6} Find the area of the region bounded by the curve, the \(x\)-axis and the lines \(x = a\) and \(x = 2a\), where \(a > 0\), giving your answer in terms of \(a\) [4 marks]

The graph of \(y = \frac{2x^3}{x^2 + 1}\) is shown for \(0 \leq x \leq 4\) Caroline is attempting to approximate the shaded area, A, under the curve using the trapezium rule by splitting the area into \(n\) trapezia.

1. When \(n = 4\)
   1. State the number of ordinates that Caroline uses. [1 mark]
   2. Calculate the area that Caroline should obtain using this method. Give your answer correct to two decimal places. [3 marks]
2. Show that the exact area of \(A\) is $$16 - \ln 17$$ Fully justify your answer. [5 marks]
3. Explain what would happen to Caroline's answer to part (a)(ii) as \(n \to \infty\) [1 mark]

The graph of \(y = x^3\) is shown. \includegraphics{figure_1} Find the total shaded area. Circle your answer. [1 mark] \(-68\) 60 68 128

The graph of \(y = f(x)\) intersects the \(x\)-axis at \((-3, 0)\), \((0, 0)\) and \((2, 0)\) as shown in the diagram below. \includegraphics{figure_2} The shaded region \(A\) has an area of 189 The shaded region \(B\) has an area of 64 Find the value of \(\int_{-3}^{2} f(x) \, dx\) Circle your answer. [1 mark] \(-253\) \(\quad\) \(-125\) \(\quad\) \(125\) \(\quad\) \(253\)

The shaded region, shown in the diagram below, is defined by $$x^2 - 7x + 7 \leq y \leq 7 - 2x$$ \includegraphics{figure_2} Identify which of the following gives the area of the shaded region. Tick (\(\checkmark\)) one box. [1 mark] \(\int (7 - 2x) \, dx - \int (x^2 - 7x + 7) \, dx\) \(\int_0^5 (x^2 - 5x) \, dx\) \(\int_0^5 (5x - x^2) \, dx\) \(\int_0^5 (x^2 - 9x + 14) \, dx\)

The graph of \(y = x^2 - 9\) is shown below. \includegraphics{figure_1} Find the area of the shaded region. Circle your answer. [1 mark] \(-18\) \quad\quad \(-6\) \quad\quad \(6\) \quad\quad \(18\)

\includegraphics{figure_5} Figure 5 shows a sketch of part of the curve \(y = 2x + \frac{8}{x^2} - 5\), \(x > 0\). The point \(A(4, \frac{7}{2})\) lies on C. The line \(l\) is the tangent to C at the point A. The region \(R\), shown shaded in figure 5 is bounded by the line \(l\), the curve C, the line with equation \(x = 1\) and the \(x\)-axis. Find the exact area of \(R\). (Solutions based entirely on graphical or numerical methods are not acceptable.)

In this question you must show detailed reasoning. The diagram shows part of the graph of \(y = x^3 - 4x\). \includegraphics{figure_3} Determine the total area enclosed by the curve and the \(x\)-axis. [6]

\includegraphics{figure_8} The diagram shows the shaded region R bounded by the curve \(y = \sqrt{3x + 4}\), the \(x\)-axis, the \(y\)-axis, and the straight line that passes through the points \((k, 0)\) and \((4, 4)\), where \(0 < k < 4\). Region R is occupied by a uniform lamina.

1. Determine, in terms of \(k\), an expression for the \(y\)-coordinate of the centre of mass of the lamina. Give your answer in the form \(\frac{a + bk}{c + dk}\), where \(a\), \(b\), \(c\) and \(d\) are integers to be determined. [6]
2. Show that the \(y\)-coordinate of the centre of mass of the lamina cannot be \(\frac{3}{2}\). [2]

The diagram below shows a curve with equation \(y = (x + 2)(x - 2)(x + 1)\). \includegraphics{figure_16} Calculate the total area of the two shaded regions. [8]

The diagram below shows a sketch of the curve \(y = f(x)\), where \(f(x) = 10x + 3x^2 - x^3\). The curve intersects the \(x\)-axis at the origin \(O\) and at the points \(A(-2, 0)\), \(B(5, 0)\). The tangent to the curve at the point \(C(2, 24)\) intersects the \(y\)-axis at the point \(D\). \includegraphics{figure_11}

1. Find the coordinates of \(D\). [5]
2. Find the area of the shaded region. [6]
3. Determine the range of values of \(x\) for which \(f(x)\) is an increasing function. [4]

The diagram below shows a sketch of the curve C with equation \(y = 2 - 3x - 2x^2\) and the line L with equation \(y = x + 2\). The curve and the line intersect the coordinate axes at the points A and B. \includegraphics{figure_14}

1. Write down the coordinates of A and B. [2]
2. Calculate the area enclosed by C and L. [6]

\includegraphics{figure_17} The diagram above shows a sketch of the curve \(y = 3x - x^2\). The curve intersects the \(x\)-axis at the origin and at the point \(A\). The tangent to the curve at the point \(B(2, 2)\) intersects the \(x\)-axis at the point \(C\).

1. Find the equation of the tangent to the curve at \(B\). [4]
2. Find the area of the shaded region. [8]

The aerial view of a patio under construction is shown below. \includegraphics{figure_9} The curved edge of the patio is described by the equation \(9x^2 + 16y^2 = 144\), where \(x\) and \(y\) are measured in metres. To construct the patio, the area enclosed by the curve and the coordinate axes is to be covered with a layer of concrete of depth 0.06 m.

1. Show that the volume of concrete required for the construction of the patio is given by \(0.015 \int_0^4 \sqrt{144 - 9x^2}\,dx\). [3]
2. Use the trapezium rule with six ordinates to estimate the volume of concrete required. [4]
3. State whether your answer in part (b) is an overestimate or an underestimate of the volume required. Give a reason for your answer. [1]

\includegraphics{figure_5} Figure 5 shows a sketch of the curve \(C\) with equation \(y = (x - 2)^2(x + 3)\) The region \(R\), shown shaded in Figure 5, is bounded by \(C\), the vertical line passing through the maximum turning point of \(C\) and the \(x\)-axis. Find the exact area of \(R\). *(Solutions based entirely on graphical or numerical methods are not acceptable.)* [6]

The diagram below shows part of a curve C with equation \(y = 1 + 3x - \frac{1}{2}x^2\). \includegraphics{figure_7}

1. The curve crosses the \(y\) axis at the point A. The straight line L is normal to the curve at A and meets the curve again at B. Find the equation of L and the \(x\) coordinate of the point B. [4]
2. The region R is bounded by the curve C and the line L. Find the exact area of R. [3]