Further Integration: exam review Exam review: second paper

Find the following. [6]

1. (a) (a) \(\int\left(9\sqrt{x}-\frac{14}{\sqrt{x}}\right)dx\) [3]
2. (b) (b) Given that \(\frac{dy}{dx}=4e^{x}-6\sin x\) and that \(y=7\) when \(x=0\), find \(y\) in terms of \(x\). [3]

Find each integral. [6]

1. (a) (a) \(\int\frac{12}{5-4x}dx\) [2]
2. (b) (b) \(\int\cos(2-5x)dx\) [2]
3. (c) (c) \(\int\frac{10x}{x^{2}+6}dx\) [2]

Find the exact value of each integral. Show your working. [6]

1. (a) (a) \(\int_{0}^{\ln 2}9e^{3x}dx\) [3]
2. (b) (b) \(\int_{-6}^{-2}\frac{9}{x}dx\) [3]

The curve \(C\) has equation \(y=x^{3}-9x^{2}+20x\). [5]

1. (a) (a) Write down the \(x\)-intercepts of \(C\). [1]
2. (b) (b) Write down an expression for the total area of the regions enclosed by \(C\) and the \(x\)-axis. [2]
3. (c) (c) Use your calculator to find the total area. [2]

Use the substitution given in each part. [6]

1. (a) (a) Use the substitution \(u=x^{2}+5\) to show that \(\int_{0}^{2}x\sqrt{x^{2}+5}\,dx=\frac{27-5\sqrt{5}}{3}\). [4]
2. (b) (b) Use the substitution \(u=2+\cos x\) to find \(\int\frac{\sin x}{2+\cos x}\,dx\). [2]

The function \(f\) satisfies \(\int_{2}^{6}f(x)\,dx=9\). [6]

1. (a) (a) Find \(\int_{2}^{6}\left(5f(x)+2\right)dx\). [2]
2. (b) (b) Find \(\int_{4}^{12}f\left(\frac{x}{2}\right)dx\). [2]
3. (c) (c) Given that \(\int_{2}^{6}\left(af(x)-3\right)dx=24\), find the value of \(a\). [2]

Find the exact value of each integral. Show your working. [6]

1. (a) (a) \(\int_0^{\ln 5}10e^{-3x}dx\) [2]
2. (b) (b) \(\int_{-6}^{-2}\frac{9}{x}dx\) [2]
3. (c) (c) \(\int_{1}^{4}\frac{5}{x-7}dx\) [2]

A function \(f\) satisfies \(f'(x)=\frac{8}{x}\) for \(x<0\) and \(f(-2)=1\). [6]

1. (a) (a) Find the exact value of \(f(-5)\). [3]
2. (b) (b) Given that \(a>0\) and \(\int_a^{2a}(x+2)dx=\frac{39}{2}\), find the value of \(a\). [3]

The graph of \(y=f(x)\) passes through the point \(P(6,3)\). The function \(g\) is defined by \(g(x)=3f\left(\frac{x}{2}\right)-1\). It is given that \(\int_{0}^{6}f(x)\,dx=4\). [5]

1. (a) (a) Describe the sequence of transformations that maps the graph of \(y=f(x)\) to the graph of \(y=g(x)\), and write down the coordinates of the image of \(P\). [2]
2. (b) (b) Find \(\int_{0}^{12}g(x)\,dx\). [3]

The diagram shows part of the curve \(C\) with equation \(y=\frac{8}{x+1}\), for \(0\leq x\leq 3\). The region \(R\) is bounded by \(C\), the \(x\)-axis, the \(y\)-axis and the line \(x=3\). The region \(S\) is bounded by \(C\), the \(y\)-axis and the lines \(y=2\) and \(y=8\). [14]

1. (a) (a) Find the area of \(R\). [3]
2. (b) (b) The region \(R\) is rotated through \(360^{\circ}\) about the \(x\)-axis. Find the exact volume of the solid formed. [3]
3. (c) (c) Find the area of \(S\). [4]
4. (d) (d) The region \(S\) is rotated through \(360^{\circ}\) about the \(y\)-axis. Find the volume of the solid formed. [4]

The diagram shows the graph of \(y=f(x)\), where \(f(x)=x^{3}-9x^{2}+18x\). The graph crosses the \(x\)-axis at \(x=0\), \(x=3\) and \(x=6\). The shaded region \(A\) lies between \(x=0\) and \(x=3\) and the shaded region \(B\) lies between \(x=3\) and \(x=6\). [12]

1. (a) (a) Find \(\int_{0}^{3}f(x)\,dx\). [2]
2. (b) (b) Find the total area of the regions \(A\) and \(B\). [3]
3. (c) (c) The function \(g\) is defined by \(g(x)=4f(x)-1\). Find \(\int_{0}^{3}g(x)\,dx\). [2]
4. (d) (d) The function \(h\) is defined by \(h(x)=f\left(\frac{x}{3}\right)\). Find \(\int_{0}^{9}h(x)\,dx\). [2]
5. (e) (e) Given that \(\int_{0}^{3}\left(kf(x)-5\right)dx=66\), find the value of \(k\). [2]
6. (f) (f) Find \(\int_{4}^{7}f(x-4)\,dx\). [1]

The region \(R\) is enclosed by the parabola \(y=x^{2}\) and the line \(y=3x\). Units are metres. [13]

1. (a) (a) Find the coordinates of the points where the line and the parabola meet. [2]
2. (b) (b) Find the area of \(R\). [2]
3. (c) (c) \(R\) is rotated through \(360^{\circ}\) about the \(x\)-axis. Find the volume of the solid formed. [3]
4. (d) (d) \(R\) is rotated through \(360^{\circ}\) about the \(y\)-axis. Find the volume of the solid formed. [4]
5. (e) (e) The part of the line \(y=3x\) from \(x=0\) to \(x=a\), \(a>0\), is rotated through \(360^{\circ}\) about the \(x\)-axis. The volume of the cone formed is \(24\pi\). Find the value of \(a\). [2]

The number of birds, \(N\) hundred, in a nature reserve \(t\) years after it opens satisfies \(\frac{dN}{dt}=\frac{12t}{t^{2}+6}\), \(t\geq 0\). Initially there are 300 birds. [13]

1. (a) (a) Use the substitution \(u=t^{2}+6\) to find \(N(t)\). [4]
2. (b) (b) Find the value of \(N\) when \(t=5\). [2]
3. (c) (c) Find the exact change in \(N\) between \(t=2\) and \(t=8\). [3]
4. (d) (d) Use your calculator to find the value of \(t\) for which \(N=20\). [2]
5. (e) (e) Find the value that \(\frac{dN}{dt}\) approaches as \(t\to\infty\). Comment on the long-term behaviour of \(N\). [2]

A pottery designs a planter. Units are decimetres, so that 1 dm\(^3\) is one litre. A planter is made by rotating the region bounded by the curve \(y=\sqrt{x^2+4}\), the \(y\)-axis and the line \(y=4\) through \(360^{\circ}\) about the \(y\)-axis. [11]

1. (a) (a) Show that \(x^2=y^2-4\). [1]
2. (b) (b) Find the exact capacity of the planter, in dm\(^3\). [3]
3. (c) (c) Write down the capacity of the planter in litres, correct to 3 significant figures. [1]
4. (d) (d) The planter is filled with 15 litres of water. Find the depth of the water, to 3 significant figures. [3]
5. (e) (e) Find the area of the region used to make the planter, giving your answer to 3 significant figures. [3]