Further Integration: exam review Exam review

Find the following. [6]

1. (a) (a) \(\int\left(6\sqrt{x}-\frac{10}{\sqrt{x}}\right)dx\) [3]
2. (b) (b) Given that \(\frac{dy}{dx}=6e^{x}-3\cos x\) and that \(y=4\) when \(x=0\), find \(y\) in terms of \(x\). [3]

Find each integral. [6]

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

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

1. (a) (a) \(\int_{0}^{\ln 3}4e^{2x}dx\) [3]
2. (b) (b) \(\int_{-4}^{-1}\frac{6}{x}dx\) [3]

The curve \(C\) has equation \(y=x^{3}-7x^{2}+10x\). [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 \(u=x^{2}+3\) to show that \(\int_{0}^{1}x\sqrt{x^{2}+3}\,dx=\frac{8-3\sqrt{3}}{3}\). [4]
2. (b) (b) Use the substitution \(u=1+\sin x\) to find \(\int\frac{\cos x}{1+\sin x}\,dx\). [2]

The function \(f\) satisfies \(\int_{1}^{5}f(x)\,dx=12\). [6]

1. (a) (a) Find \(\int_{1}^{5}\left(4f(x)+3\right)dx\). [2]
2. (b) (b) Find \(\int_{2}^{10}f\left(\frac{x}{2}\right)dx\). [2]
3. (c) (c) Given that \(\int_{1}^{5}\left(af(x)-2\right)dx=28\), find the value of \(a\). [2]

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

1. (a) (a) \(\int_0^{\ln 3}6e^{-2x}dx\) [2]
2. (b) (b) \(\int_{-4}^{-1}\frac{6}{x}dx\) [2]
3. (c) (c) \(\int_{0}^{3}\frac{6}{x-5}dx\) [2]

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

1. (a) (a) Find the exact value of \(f(-3)\). [3]
2. (b) (b) Given that \(a>0\) and \(\int_a^{2a}(x+1)dx=8\), find the value of \(a\). [3]

The graph of \(y=f(x)\) passes through the point \(P(4,-2)\). The function \(g\) is defined by \(g(x)=2f\left(\frac{x}{3}\right)+1\). It is given that \(\int_{0}^{4}f(x)\,dx=5\). [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 \(y=\frac{6}{x+2}\), for \(0\leq x\leq 4\). The region \(R\) is bounded by \(C\), the \(x\)-axis, the \(y\)-axis and the line \(x=4\). The region \(S\) is bounded by \(C\), the \(y\)-axis and the lines \(y=1\) and \(y=3\). [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 \(f(x)=x^{3}-6x^{2}+8x\). The graph crosses the \(x\)-axis at \(x=0\), \(x=2\) and \(x=4\). The shaded region \(A\) lies between \(x=0\) and \(x=2\) and the shaded region \(B\) lies between \(x=2\) and \(x=4\). [12]

1. (a) (a) Find \(\int_{0}^{2}f(x)\,dx\). [2]
2. (b) (b) Find the total area of the regions \(A\) and \(B\). [3]
3. (c) (c) The function \(g\) is defined by \(g(x)=3f(x)+2\). Find \(\int_{0}^{2}g(x)\,dx\). [2]
4. (d) (d) The function \(h\) is defined by \(h(x)=f\left(\frac{x}{2}\right)\). Find \(\int_{0}^{4}h(x)\,dx\). [2]
5. (e) (e) Given that \(\int_{0}^{2}\left(kf(x)-3\right)dx=10\), find the value of \(k\). [2]
6. (f) (f) Find \(\int_{3}^{5}f(x-3)\,dx\). [1]

The region \(R\) is enclosed by the parabola \(y=x^{2}\) and the line \(y=2x\). Units are centimetres. [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 \(y=2x\) from \(x=0\) to \(x=a\), \(a>0\), is rotated through \(360^{\circ}\) about the \(x\)-axis. The volume of the cone formed is \(36\pi\). Find the value of \(a\). [2]

The concentration, \(C\) mg per litre, of a drug in a patient's blood \(t\) hours after an injection satisfies \(\frac{dC}{dt}=\frac{6t}{t^{2}+4}\), \(t\geq 0\). Initially the concentration is 2 mg per litre. [13]

1. (a) (a) Use the substitution \(u=t^{2}+4\) to find \(C(t)\). [4]
2. (b) (b) Find the value of \(C\) when \(t=6\). [2]
3. (c) (c) Find the exact change in \(C\) between \(t=2\) and \(t=6\). [3]
4. (d) (d) Use your calculator to find the value of \(t\) for which \(C=10\). [2]
5. (e) (e) Find the value that \(\frac{dC}{dt}\) approaches as \(t\to\infty\). Comment on the long-term behaviour of \(C\). [2]

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

1. (a) (a) Show that \(x^2=y^2-1\). [1]
2. (b) (b) Find the exact capacity of the vase, in dm\(^3\). [3]
3. (c) (c) Write down the capacity of the vase in litres, correct to 3 significant figures. [1]
4. (d) (d) The vase is filled with 10 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 vase, giving your answer to 3 significant figures. [3]