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

The finite region \(R\) is bounded by the curve \(y = 1 + 3\sqrt{x}\), the \(x\)-axis and the lines \(x = 2\) and \(x = 8\).

1. Use the trapezium rule with three intervals of equal width to estimate to 3 significant figures the area of \(R\). [6]
2. Use integration to find the exact area of \(R\) in the form \(a + b\sqrt{2}\). [5]
3. Find the percentage error in the estimate made in part (a). [2]

\includegraphics{figure_2} Figure 2 shows the curve with equation \(y = 5 + x - x^2\) and the normal to the curve at the point \(P(1, 5)\).

1. Find an equation for the normal to the curve at \(P\) in the form \(y = mx + c\). [5]
2. Find the coordinates of the point \(Q\), where the normal to the curve at \(P\) intersects the curve again. [2]
3. Show that the area of the shaded region bounded by the curve and the straight line \(PQ\) is \(\frac{4}{3}\). [6]

\includegraphics{figure_3} Figure 3 shows part of the curve \(y = \text{f}(x)\) where $$\text{f}(x) = \frac{1 - 8x^3}{x^2}, \quad x \neq 0.$$

1. Solve the equation \(\text{f}(x) = 0\). [3]
2. Find \(\int \text{f}(x) \, dx\). [3]
3. Find the area of the shaded region bounded by the curve \(y = \text{f}(x)\), the \(x\)-axis and the line \(x = 2\). [3]

\includegraphics{figure_2} Figure 2 shows the curve \(C\) with equation \(y = 3x - 4\sqrt{x} + 2\) and the tangent to \(C\) at the point \(A\). Given that \(A\) has \(x\)-coordinate 4,

1. show that the tangent to \(C\) at \(A\) has the equation \(y = 2x - 2\). [6]

The shaded region is bounded by \(C\), the tangent to \(C\) at \(A\) and the positive coordinate axes.

1. Find the area of the shaded region. [8]

\includegraphics{figure_9} The diagram shows the curve \(y = 2x^2 + 6x + 7\) and the straight line \(y = 2x + 13\).

1. Find the coordinates of the points where the curve and line intersect. [4]
2. Show that the area of the shaded region bounded by the curve and line is given by $$\int_{-3}^{1} (6 - 4x - 2x^2) dx.$$ [2]
3. Hence find the area of the shaded region. [5]

\includegraphics{figure_6} The diagram shows the curve with equation \(y = 4x^{\frac{1}{3}} - x\), \(x \geq 0\). The curve meets the \(x\)-axis at the origin and at the point \(A\) with coordinates \((a, 0)\).

1. Show that \(a = 8\). [3]
2. Find the area of the finite region bounded by the curve and the positive \(x\)-axis. [5]

Find the area of the finite region enclosed by the curve \(y = 5x - x^2\) and the \(x\)-axis. [6]

\includegraphics{figure_1} Fig. 9 shows a sketch of the graph of \(y = x^3 - 10x^2 + 12x + 72\).

1. Write down \(\frac{dy}{dx}\). [2]
2. Find the equation of the tangent to the curve at the point on the curve where \(x = 2\). [4]
3. Show that the curve crosses the \(x\)-axis at \(x = -2\). Show also that the curve touches the \(x\)-axis at \(x = 6\). [3]
4. Find the area of the finite region bounded by the curve and the \(x\)-axis, shown shaded in Fig. 9. [4]

Fig. 10 shows a sketch of the curve \(y = x^2 - 4x + 3\). The point A on the curve has \(x\)-coordinate 4. At point B the curve crosses the \(x\)-axis. \includegraphics{figure_2}

1. Use calculus to find the equation of the normal to the curve at A and show that this normal intersects the \(x\)-axis at C (16, 0). [6]
2. Find the area of the region ABC bounded by the curve, the normal at A and the \(x\)-axis. [5]

\includegraphics{figure_6} The diagram shows the curve with equation \(y = \frac{1}{\sqrt{3x + 2}}\). The shaded region is bounded by the curve and the lines \(x = 0\), \(x = 2\) and \(y = 0\).

1. Find the exact area of the shaded region. [4]
2. The shaded region is rotated completely about the \(x\)-axis. Find the exact volume of the solid formed, simplifying your answer. [5]

\includegraphics{figure_4} The diagram shows part of the curve \(y = \frac{k}{x}\), where \(k\) is a positive constant. The points A and B on the curve have \(x\)-coordinates 2 and 6 respectively. Lines through A and B parallel to the axes as shown meet at the point C. The region R is bounded by the curve and the lines \(x = 2\), \(x = 6\) and \(y = 0\). The region S is bounded by the curve and the lines AC and BC. It is given that the area of the region R is \(\ln 81\).

1. Show that \(k = 4\). [3]
2. Find the exact volume of the solid produced when the region S is rotated completely about the \(x\)-axis. [4]

\includegraphics{figure_7} The diagram shows the curve with equation \(y = (3x - 1)^4\). The point P on the curve has coordinates \((1, 16)\) and the tangent to the curve at P meets the \(x\)-axis at the point Q. The shaded region is bounded by PQ, the \(x\)-axis and that part of the curve for which \(\frac{1}{3} \leqslant x \leqslant 1\). Find the exact area of this shaded region. [10]

Fig. 9 shows the line \(y = x\) and the curve \(y = f(x)\), where \(f(x) = \frac{1}{2}(e^x - 1)\). The line and the curve intersect at the origin and at the point P\((a, a)\). \includegraphics{figure_9}

1. Show that \(e^a = 1 + 2a\). [1]
2. Show that the area of the region enclosed by the curve, the \(x\)-axis and the line \(x = a\) is \(\frac{1}{2}a\). Hence find, in terms of \(a\), the area enclosed by the curve and the line \(y = x\). [6]
3. Show that the inverse function of f\((x)\) is g\((x)\), where g\((x) = \ln(1 + 2x)\). Add a sketch of \(y = g(x)\) to the copy of Fig. 9. [5]
4. Find the derivatives of f\((x)\) and g\((x)\). Hence verify that \(g'(a) = \frac{1}{f'(a)}\). Give a geometrical interpretation of this result. [7]

Fig. 9 shows the curve \(y = xe^{-2x}\) together with the straight line \(y = mx\), where \(m\) is a constant, with \(0 < m < 1\). The curve and the line meet at O and P. The dashed line is the tangent at P. \includegraphics{figure_9}

1. Show that the \(x\)-coordinate of P is \(-\frac{1}{2}\ln m\). [3]
2. Find, in terms of \(m\), the gradient of the tangent to the curve at P. [4]

You are given that OP and this tangent are equally inclined to the \(x\)-axis.

1. Show that \(m = e^{-2}\), and find the exact coordinates of P. [4]
2. Find the exact area of the shaded region between the line OP and the curve. [7]

\includegraphics{figure_7} The diagram shows the curve with equation \(y = 2x - e^{\frac{1}{2}x}\). The shaded region is bounded by the curve, the \(x\)-axis and the lines \(x = 2\) and \(x = 4\).

1. Find the area of the shaded region, giving your answer in terms of e. [4]

The shaded region is rotated through four right angles about the \(x\)-axis.

1. Using Simpson's rule with two strips, estimate the volume of the solid formed. [5]

\includegraphics{figure_3} The curve \(C\) with equation \(y = 2e^x + 5\) meets the \(y\)-axis at the point \(M\), as shown in Fig. 3.

1. Find the equation of the normal to \(C\) at \(M\) in the form \(ax + by = c\), where \(a\), \(b\) and \(c\) are integers. [4]

This normal to \(C\) at \(M\) crosses the \(x\)-axis at the point \(N(n, 0)\).

1. Show that \(n = 14\). [1]

The point \(P(\ln 4, 13)\) lies on \(C\). The finite region \(R\) is bounded by \(C\), the axes and the line \(PN\), as shown in Fig. 3.

1. Find the area of \(R\), giving your answers in the form \(p + q \ln 2\), where \(p\) and \(q\) are integers to be found. [7]

\includegraphics{figure_2} Figure 2 shows the curve with parametric equations $$x = t + \sin t, \quad y = \sin t, \quad 0 \leq t \leq \pi.$$

1. Find \(\frac{dy}{dx}\) in terms of \(t\). [3]
2. Find, in exact form, the coordinates of the point where the tangent to the curve is parallel to the \(x\)-axis. [3]
3. Show that the region bounded by the curve and the \(x\)-axis has area 2. [6]

\includegraphics{figure_1} Figure 1 shows a sketch of part of the curve with equation $$y = \sin (\cos x).$$ The curve cuts the \(x\)-axis at the points \(A\) and \(C\) and the \(y\)-axis at the point \(B\).

1. Find the coordinates of the points \(A\), \(B\) and \(C\). [3]
2. Prove that \(B\) is a stationary point. [2]

Given that the region \(OCB\) is convex,

1. show that, for \(0 \leq x \leq \frac{\pi}{2}\), $$\sin (\cos x) \leq \cos x$$ and $$(1 - \frac{2}{\pi} x) \sin 1 \leq \sin (\cos x)$$ and state in each case the value or values of \(x\) for which equality is achieved. [6]
2. Hence show that $$\frac{\pi}{4} \sin 1 < \int_0^{\frac{\pi}{2}} \sin(\cos x) \, dx < 1.$$ [4]

\includegraphics{figure_2} Figure 2 shows a sketch of part of two curves \(C_1\) and \(C_2\) for \(y \geq 0\). The equation of \(C_1\) is \(y = m_1 - x^{n_1}\) and the equation of \(C_2\) is \(y = m_2 - x^{n_2}\), where \(m_1\), \(m_2\), \(n_1\) and \(n_2\) are positive integers with \(m_2 > m_1\). Both \(C_1\) and \(C_2\) are symmetric about the line \(x = 0\) and they both pass through the points \((3, 0)\) and \((-3, 0)\). Given that \(n_1 + n_2 = 12\), find

1. the possible values of \(n_1\) and \(n_2\), [4]
2. the exact value of the smallest possible area between \(C_1\) and \(C_2\), simplifying your answer, [8]
3. the largest value of \(x\) for which the gradients of the two curves can be the same. Leave your answer in surd form. [5]

$$f(x) = x - [x], \quad x \geq 0$$ where \([x]\) is the largest integer \(\leq x\). For example, \(f(3.7) = 3.7 - 3 = 0.7\); \(f(3) = 3 - 3 = 0\).

1. Sketch the graph of \(y = f(x)\) for \(0 \leq x < 4\). [3]
2. Find the value of \(p\) for which \(\int_2^p f(x) dx = 0.18\). [3]

Given that $$g(x) = \frac{1}{1+kx}, \quad x \geq 0, \quad k > 0,$$ and that \(x_0 = \frac{1}{2}\) is a root of the equation \(f(x) = g(x)\),

1. find the value of \(k\). [2]
2. Add a sketch of the graph of \(y = g(x)\) to your answer to part \((a)\). [1]

The root of \(f(x) = g(x)\) in the interval \(n < x < n + 1\) is \(x_n\), where \(n\) is an integer.

1. Prove that $$2 x_n^2 - (2n - 1)x_n - (n + 1) = 0.$$ [4]
2. Find the smallest value of \(n\) for which \(x_n - n < 0.05\). [4]

\includegraphics{figure_1} Figure 1 shows a sketch of the curve \(C\) with equation $$y = \cos x \ln(\sec x), \quad -\frac{\pi}{2} < x < \frac{\pi}{2}$$ The points \(A\) and \(B\) are maximum points on \(C\).

1. Find the coordinates of \(B\) in terms of e. [5]

The finite region \(R\) lies between \(C\) and the line \(AB\).

1. Show that the area of \(R\) is $$\frac{2}{e} \arccos \left(\frac{1}{e}\right) + 2\ln \left(e + \sqrt{(e^2 - 1)}\right) - \frac{4}{e} \sqrt{(e^2 - 1)}.$$ [arccos \(x\) is an alternative notation for \(\cos^{-1} x\)] [8]

In this question you must show detailed reasoning. Find the exact area of the region enclosed by the curve \(y = \frac{1}{x+2}\), the two axes and the line \(x = 2.5\). [3]

In this question you must show detailed reasoning. The function f is defined by \(\text{f}(x) = \cos x + \sqrt{3} \sin x\) with domain \(0 \leqslant x \leqslant 2\pi\).

1. Solve the following equations.
   1. \(\text{f}'(x) = 0\) [4]
   2. \(\text{f}''(x) = 0\) [3]
   The diagram shows the graph of the gradient function \(y = \text{f}'(x)\) for the domain \(0 \leqslant x \leqslant 2\pi\). \includegraphics{figure_5}
2. Use your answers to parts (a)(i) and (a)(ii) to find the coordinates of points \(A\), \(B\), \(C\) and \(D\). [2]
3. 1. Explain how to use the graph of the gradient function to find the values of \(x\) for which f(x) is increasing. [1]
   2. Using set notation, write down the set of values of \(x\) for which f(x) is increasing in the domain \(0 \leqslant x \leqslant 2\pi\). [2]

\includegraphics{figure_5} The diagram shows the curve \(C\) with parametric equations \(x = \frac{3}{t}\), \(y = t^2 e^{-2t}\), where \(t > 0\). The maximum point on \(C\) is denoted by \(P\).

1. Determine the exact coordinates of \(P\). [4] The shaded region \(R\) is enclosed by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 6\).
2. Show that the area of \(R\) is given by $$\int_a^b 3te^{-2t} dt,$$ where \(a\) and \(b\) are constants to be determined. [3]
3. Hence determine the exact area of \(R\). [5]