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

\includegraphics{figure_3} Figure 3 shows the shaded region \(R\) which is bounded by the curve \(y = -2x^2 + 4x\) and the line \(y = \frac{3}{2}\). The points \(A\) and \(B\) are the points of intersection of the line and the curve. Find

1. the \(x\)-coordinates of the points \(A\) and \(B\), [4]
2. the exact area of \(R\). [6]

The curve \(C\) has equation \(y = x\sqrt{x^2 + 1}, \quad 0 \leq x \leq 2\).

1. Copy and complete the table below, giving the values of \(y\) to 3 decimal places at \(x = 1\) and \(x = 1.5\).

   \(x\)	0	0.5	1	1.5	2
   \(y\)	0	0.530			6

   [2]
2. Use the trapezium rule, with all the \(y\) values from your table, to find an approximation for the value of \(\int_0^2 x\sqrt{x^2 + 1} \, dx\), giving your answer to 3 significant figures. [4]

\includegraphics{figure_2} Figure 2 shows the curve \(C\) with equation \(y = x\sqrt{x^2 + 1}\), \(0 \leq x \leq 2\), and the straight line segment \(l\), which joins the origin and the point \((2, 6)\). The finite region \(R\) is bounded by \(C\) and \(l\).

1. Use your answer to part (b) to find an approximation for the area of \(R\), giving your answer to 3 significant figures. [3]

\includegraphics{figure_2} In Figure 2 the curve \(C\) has equation \(y = 6x - x^2\) and the line \(L\) has equation \(y = 2x\).

1. Show that the curve \(C\) intersects the \(x\)-axis at \(x = 0\) and \(x = 6\). [1]
2. Show that the line \(L\) intersects the curve \(C\) at the points \((0, 0)\) and \((4, 8)\). [3]

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

1. Use calculus to find the area of \(R\). [6]

\includegraphics{figure_2} Figure 2 shows part of the curve \(C\) with equation $$y = 9 - 2x - \frac{2}{\sqrt{x}}, \quad x > 0.$$ The point \(A(1, 5)\) lies on \(C\) and the curve crosses the \(x\)-axis at \(B(b, 0)\), where \(b\) is a constant and \(b > 0\).

1. Verify that \(b = 4\). [1]

The tangent to \(C\) at the point \(A\) cuts the \(x\)-axis at the point \(D\), as shown in Fig. 2.

1. Show that an equation of the tangent to \(C\) at \(A\) is \(y + x = 6\). [4]
2. Find the coordinates of the point \(D\). [1]

The shaded region \(R\), shown in Fig. 2, is bounded by \(C\), the line \(AD\) and the \(x\)-axis.

1. Use integration to find the area of \(R\). [6]

\includegraphics{figure_1} Figure 1 shows the curve with equation \(y = 5 + 2x - x^2\) and the line with equation \(y = 2\). The curve and the line intersect at the points \(A\) and \(B\).

1. Find the \(x\)-coordinates of \(A\) and \(B\). [3]

The shaded region \(R\) is bounded by the curve and the line.

1. Find the area of \(R\). [6]

\includegraphics{figure_5} Figure 2 shows part of the curve with equation $$y = x^3 - 6x^2 + 9x.$$ The curve touches the \(x\)-axis at \(A\) and has a maximum turning point at \(B\).

1. Show that the equation of the curve may be written as $$y = x(x - 3)^2,$$ and hence write down the coordinates of \(A\). [2]
2. Find the coordinates of \(B\). [5]

The shaded region \(R\) is bounded by the curve and the \(x\)-axis.

1. Find the area of \(R\). [5]

\includegraphics{figure_7} Figure 2 shows the line with equation \(y = x + 1\) and the curve with equation \(y = 6x - x^2 - 3\). The line and the curve intersect at the points \(A\) and \(B\), and \(O\) is the origin.

1. Calculate the coordinates of \(A\) and the coordinates of \(B\). [5]

The shaded region \(R\) is bounded by the line and the curve.

1. Calculate the area of \(R\). [7]

\includegraphics{figure_11} The curve \(C\), shown in Fig. 2, represents the graph of $$y = \frac{x^2}{25}, \quad x \geq 0.$$ The points \(A\) and \(B\) on the curve \(C\) have \(x\)-coordinates 5 and 10 respectively.

1. Write down the \(y\)-coordinates of \(A\) and \(B\). [1]
2. Find an equation of the tangent to \(C\) at \(A\). [4]

The finite region \(R\) is enclosed by \(C\), the \(y\)-axis and the lines through \(A\) and \(B\) parallel to the \(x\)-axis.

1. For points \((x, y)\) on \(C\), express \(x\) in terms of \(y\). [2]
2. Use integration to find the area of \(R\). [5]

\includegraphics{figure_2} Figure 2 shows a sketch of the curve defined by the parametric equations $$x = t^2 + 2t \quad y = \frac{2}{t(3-t)} \quad a \leq t \leq b$$ where \(a\) and \(b\) are constants. The ends of the curve lie on the line with equation \(y = 1\)

1. Find the value of \(a\) and the value of \(b\) [2]

The region \(R\), shown shaded in Figure 2, is bounded by the curve and the line with equation \(y = 1\)

1. Show that the area of region \(R\) is given by $$M - k \int_a^b \frac{t+1}{t(3-t)} dt$$ where \(M\) and \(k\) are constants to be found. [5]
2. 1. Write \(\frac{t+1}{t(3-t)}\) in partial fractions.
   2. Use algebraic integration to find the exact area of \(R\), giving your answer in simplest form. [6]

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. \includegraphics{figure_2} Figure 2 shows a sketch of part of the curve with equation $$y = \frac{12\sqrt{x}}{(2x^2 + 3)^3}$$ The region \(R\), shown shaded in Figure 2, is bounded by the curve, the line with equation \(x = \frac{1}{\sqrt{2}}\), the \(x\)-axis and the line with equation \(x = k\). This region is rotated through \(360°\) about the \(x\)-axis to form a solid of revolution. Given that the volume of this solid is \(\frac{713\pi}{648}\), use algebraic integration to find the exact value of the constant \(k\). [6]

\includegraphics{figure_1} Figure 1 shows a sketch of part of the curve with equation \(y = 4x - xe^{\frac{1}{x}}, x \geqslant 0\) The curve meets the \(x\)-axis at the origin \(O\) and cuts the \(x\)-axis at the point \(A\).

1. Find, in terms of \(\ln 2\), the \(x\) coordinate of the point \(A\). [2]
2. Find $$\int xe^{\frac{1}{x}} dx$$ [3]
3. Find, by integration, the exact value for the area of \(R\). Give your answer in terms of \(\ln 2\) [3]

The finite region \(R\), shown shaded in Figure 1, is bounded by the \(x\)-axis and the curve with equation $$y = 4x - xe^{\frac{1}{x}}, x \geqslant 0$$

\includegraphics{figure_2} Part of the design of a stained glass window is shown in Fig. 2. The two loops enclose an area of blue glass. The remaining area within the rectangle \(ABCD\) is red glass. The loops are described by the curve with parametric equations $$x = 3 \cos t, \quad y = 9 \sin 2t, \quad 0 \leq t < 2\pi.$$

1. Find the cartesian equation of the curve in the form \(y^2 = f(x)\). [4]
2. Show that the shaded area in Fig. 2, enclosed by the curve and the \(x\)-axis, is given by $$\int_0^{\frac{\pi}{2}} A \sin 2t \sin t \, dt, \text{ stating the value of the constant } A.$$ [3]
3. Find the value of this integral. [4]

The sides of the rectangle \(ABCD\), in Fig. 2, are the tangents to the curve that are parallel to the coordinate axes. Given that 1 unit on each axis represents 1 cm,

1. find the total area of the red glass. [4]

The rudder on a ship is modelled as a uniform plane lamina having the same shape as the region \(R\) which is enclosed between the curve with equation \(y = 2x - x^2\) and the \(x\)-axis.

1. Show that the area of \(R\) is \(\frac{4}{3}\). [4]
2. Find the coordinates of the centre of mass of the lamina. [5]

\includegraphics{figure_2} The curve \(C\), shown in Fig. 2, represents the graph of $$y = \frac{x^2}{25}, x \geq 0.$$ The points \(A\) and \(B\) on the curve \(C\) have \(x\)-coordinates \(5\) and \(10\) respectively.

1. Write down the \(y\)-coordinates of \(A\) and \(B\). [1]
2. Find an equation of the tangent to \(C\) at \(A\). [4]

The finite region \(R\) is enclosed by \(C\), the \(y\)-axis and the lines through \(A\) and \(B\) parallel to the \(x\)-axis.

1. For points \((x, y)\) on \(C\), express \(x\) in terms of \(y\). [2]
2. Use integration to find the area of \(R\). [5]

Figure 2 \includegraphics{figure_2} Figure 2 shows part of the curve with equation $$y = x³ - 6x² + 9x.$$ The curve touches the x-axis at A and has a maximum turning point at B.

1. Show that the equation of the curve may be written as $$y = x(x - 3)²,$$ and hence write down the coordinates of A. [2]
2. Find the coordinates of B. [5]

The shaded region R is bounded by the curve and the x-axis.

1. Find the area of R. [5]

\includegraphics{figure_2} Figure 1 shows the curve with equation \(y = 5 + 2x - x^2\) and the line with equation \(y = 2\). The curve and the line intersect at the points A and B.

1. Find the x-coordinates of A and B. [3]

The shaded region R is bounded by the curve and the line.

1. Find the area of R. [6]

\includegraphics{figure_3} Figure 3 shows part of the curve \(C\) with equation $$y = \frac{3}{5}x^2 - \frac{1}{4}x^3.$$ The curve \(C\) touches the \(x\)-axis at the origin and passes through the point \(A(p, 0)\).

1. Show that \(p = 6\). [1]
2. Find an equation of the tangent to \(C\) at \(A\). [4]

The curve \(C\) has a maximum at the point \(P\).

1. Find the \(x\)-coordinate of \(P\). [2]

The shaded region \(R\), in Fig. 3, is bounded by \(C\) and the \(x\)-axis.

1. Find the area of \(R\). [4]

\includegraphics{figure_2} Figure 2 shows part of the curve C with equation y = f(x), where $$f(x) = x^3 - 6x^2 + 5x.$$ The curve crosses the x-axis at the origin O and at the points A and B.

1. Factorise f(x) completely [3 marks]
2. Write down the x-coordinates of the points A and B. [1 marks]
3. Find the gradient of C at A. [3 marks] The region R is bounded by C and the line OA, and the region S is bounded by C and the line AB.
4. Use integration to find the area of the combined regions R and S, shown shaded in Fig. 2. [7 marks]

\includegraphics{figure_1} The curve \(C\), shown in Fig. 1, represents the graph of \(y = \frac{x^2}{25}\), \(x \geq 0\). The points \(A\) and \(B\) on the curve \(C\) have \(x\)-coordinates \(5\) and \(10\) respectively.

1. Write down the \(y\)-coordinates of \(A\) and \(B\). [1]
2. Find an equation of the tangent to \(C\) at \(A\). [4]

The finite region \(R\) is enclosed by \(C\), the \(y\)-axis and the lines through \(A\) and \(B\) parallel to the \(x\)-axis.

1. For points \((x, y)\) on \(C\), express \(x\) in terms of \(y\). [2]
2. Use integration to find the area of \(R\). [5]

\includegraphics{figure_1} Fig. 1 shows part of the curve \(C\) with equation \(y = \frac{1}{3}x^2 - \frac{1}{4}x^3\). The curve \(C\) touches the \(x\)-axis at the origin and passes through the point \(A(p, 0)\).

1. Show that \(p = 6\). [1]
2. Find an equation of the tangent to \(C\) at \(A\). [4]

The curve \(C\) has a maximum at the point \(P\).

1. Find the \(x\)-coordinate of \(P\). [2]

The shaded region \(R\), in Fig. 1, is bounded by \(C\) and the \(x\)-axis.

1. Find the area of \(R\). [4]

\includegraphics{figure_10} The diagram shows the graph of \(y = 1 - 3x^{-\frac{1}{2}}\).

1. Verify that the curve intersects the \(x\)-axis at \((9, 0)\). [1]
2. The shaded region is enclosed by the curve, the \(x\)-axis and the line \(x = a\) (where \(a > 9\)). Given that the area of the shaded region is 4 square units, find the value of \(a\). [9]

\includegraphics{figure_7} The diagram shows the curves \(y = -3x^2 - 9x + 30\) and \(y = x^2 + 3x - 10\).

1. Verify that the curves intersect at the points \(A(-5, 0)\) and \(B(2, 0)\). [2]
2. Show that the area of the shaded region between the curves is given by \(\int_{-5}^{2} (-4x^2 - 12x + 40) dx\). [2]
3. Hence or otherwise show that the area of the shaded region between the curves is \(228\frac{2}{3}\). [5]

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_10}

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]

A cubic curve has equation \(y = x^3 - 3x^2 + 1\).

1. Use calculus to find the coordinates of the turning points on this curve. Determine the nature of these turning points. [5]
2. Show that the tangent to the curve at the point where \(x = -1\) has gradient 9. Find the coordinates of the other point, P, on the curve at which the tangent has gradient 9 and find the equation of the normal to the curve at P. Show that the area of the triangle bounded by the normal at P and the \(x\)- and \(y\)-axes is 8 square units. [8]

Fig. 9 shows a sketch of the curve \(y = x^3 - 3x^2 - 22x + 24\) and the line \(y = 6x + 24\). \includegraphics{figure_9}

1. Differentiate \(y = x^3 - 3x^2 - 22x + 24\) and hence find the \(x\)-coordinates of the turning points of the curve. Give your answers to 2 decimal places. [4]
2. You are given that the line and the curve intersect when \(x = 0\) and when \(x = -4\). Find algebraically the \(x\)-coordinate of the other point of intersection. [3]
3. Use calculus to find the area of the region bounded by the curve and the line \(y = 6x + 24\) for \(-4 \leq x \leq 0\), shown shaded on Fig. 9. [4]