Integration: exam review Exam review

The graph shows the speed of a tram during a journey of 26 s. It speeds up steadily from rest for the first 8 s, then travels at 12 m/s until 20 s, then slows steadily to rest. [5]

1. (a) Find the distance travelled in the first 8 s. [2]
2. (b) Find the total distance travelled. [3]

Find each indefinite integral. [7]

1. (a) \(\displaystyle\int\left(9x^{2} - \frac{8}{x^{3}} + 4\right)dx\) [3]
2. (b) \(\displaystyle\int (x+3)(2x-5)\,dx\) [2]
3. (c) \(\displaystyle\int \frac{6x^{4}-9x^{2}+2}{x^{2}}\,dx\) [2]

Show your working. [7]

1. (a) Find \(\displaystyle\int \frac{5}{x^{3}}\,dx\). [2]
2. (b) Expand \(\left(x^{2}-\dfrac{3}{x^{2}}\right)^{2}\) and hence find \(\displaystyle\int \left(x^{2}-\frac{3}{x^{2}}\right)^{2}dx\). [3]
3. (c) Given that \(\displaystyle\int f(x)\,dx=2x^{3}+\frac{5}{x^{2}}+c\), find \(f(x)\). [2]

A path across a hillside is modelled by a curve, with \(x\) and \(y\) in metres. The gradient of the curve is \(\dfrac{dy}{dx}=6x^{2} + 2 - \frac{8}{x^{3}}\). The curve passes through the point \((2, 17)\). [5]

1. (a) Find the equation of the curve, in the form \(y=f(x)\). [4]
2. (b) Find the value of \(y\) on the curve when \(x=1\). [1]

A function \(f\) has gradient function \(f'(x)=3x^{2}-2x\), and \(f(0)=5\). [5]

1. (a) Find \(f(x)\). [3]
2. (b) Hence find \(\displaystyle\int_0^{2} f(x)\,dx\). [2]

The curve \(y=x^{2}-6x+10\) is shown. The region under the curve between \(x=1\) and \(x=4\) is shaded. [6]

1. (a) Write down a definite integral for the area of the shaded region. [2]
2. (b) Use your calculator to evaluate the integral. Give your answer to 3 significant figures. [2]
3. (c) Use your calculator to evaluate \(\displaystyle\int_{1}^{4}\left(2x + \frac{6}{x^{2}}\right)dx\). Give your answer to 3 significant figures. [2]

For the first 40 s after a valve opens, a sprayer releases fertiliser at a rate directly proportional to the time since it opened. After 5 s the rate is 30 grams per second. [4]

1. (a) Find an expression for the rate \(r\) grams per second, at time \(t\) seconds. [2]
2. (b) Find the amount of fertiliser released in the first 40 seconds. [2]

The rate of growth of a population of bacteria is modelled by \(y=3e^{x}\) thousand per hour, where \(x\) is the time in hours. The total change from \(x=0\) to \(x=2\) is \(\displaystyle\int_{0}^{2} 3e^{x}\,dx\). Use the trapezium rule with 4 strips to estimate this total. [7]

1. (a) Write down the values of \(y=3e^{x}\) at \(x=0.5\), \(x=1\) and \(x=1.5\), correct to 4 significant figures. [2]
2. (b) Use the trapezium rule to find the estimate. Give your answer to 3 significant figures. [3]
3. (c) State whether the estimate is greater or less than the exact value of the integral. Give a reason. [2]

The map shows a lake. The top and bottom edges of the lake are marked at equal intervals along the \(x\)-axis. Points on the top edge are (0, 62), (15, 80), (30, 91), (45, 77), (60, 58). Points on the bottom edge are (0, 14), (15, 9), (30, 16), (45, 7), (60, 12). Distances are in metres. [5]

1. (a) Use the trapezium rule to estimate the area between the top edge and the x-axis, from \(x=0\) to \(x=60\). [2]
2. (b) Use the trapezium rule to estimate the area between the bottom edge and the x-axis, over the same interval. [2]
3. (c) Hence estimate the area of the lake. Give your answer to 3 significant figures. [1]

Water is pumped into a paddling pool at a rate of \(r\) litres per minute, where \(r=kt+0.8\) and \(t\) is the time in minutes after the pump is switched on, for \(0\le t\le 10\). The graph of \(r\) against \(t\) is shown. When \(t=5\), \(r=3.3\). At \(t=0\) the pool already holds 20 litres. [12]

1. (a) Find the value of \(k\). [2]
2. (b) Find the amount in the pool when \(t=10\). [3]
3. (c) Find the value of \(t\) when the pool holds 40 litres. [3]
4. (d) The shaded region on the graph of \(r\) is the area between \(t=4\) and \(t=8\). Write down what this area represents, and find its value. [3]
5. (e) Give one criticism of this model. [1]

A tram leaves a station. The speed, \(v\) m/s, is modelled by \(v=5\sqrt{t}\) for \(0\le t\le 16\), where \(t\) is the time in seconds. Values of \(v\) are given to 4 significant figures. Time \(t\) (s): 0, 4, 8, 12, 16. Speed \(v\) (m/s): 0, 10, 14.14, 17.32, 20. [12]

1. (a) State what the area under the graph of \(v\) against \(t\) represents. [1]
2. (b) Use the trapezium rule with 4 strips and the values in the table to estimate the distance travelled in the 16 seconds. Give your answer to 3 significant figures. [3]
3. (c) Is your estimate greater or less than the true distance? Give a reason from the shape of the graph. [2]
4. (d) Use your calculator to find \(\displaystyle\int_0^{16} 5\sqrt{t}\,dt\). Give your answer to 3 significant figures. [2]
5. (e) Find the percentage error of the trapezium estimate. Give your answer to 3 significant figures. [2]
6. (f) Find the time at which the tram has travelled 100 m from the start. Give your answer to 3 significant figures. [2]

A pond is dug in a garden. Its boundary is part of the curve \(y=8x-x^{2}\) and part of the line \(y=2x\), with \(x\) and \(y\) in metres. The pond is the region shaded in the diagram. [12]

1. (a) Find the \(x\)-coordinate of the point where the curve meets the \(x\)-axis, other than the origin. [1]
2. (b) Find the \(x\)-coordinate of the point P, where the curve meets the line, other than the origin. [2]
3. (c) Write down an integral for the area of the shaded region. [2]
4. (d) Use your calculator to find the area of the shaded region. [2]
5. (e) Find the area of the region between the curve and the \(x\)-axis. [2]
6. (f) Find the percentage of the area in part (e) that the shaded region makes up. Give your answer to 3 significant figures. [2]
7. (g) The shaded region has a constant depth of 1.5 m. Find its volume. [1]

A flower bed is the region between the curve \(y=\dfrac{20}{x}\) and the line \(y=11-x\), with \(x\) and \(y\) in metres. The curve and the line meet at the points P and Q. [12]

1. (a) Use your calculator to find the \(x\)-coordinates of P and Q. [2]
2. (b) Write down an integral for the area of the flower bed, and use your calculator to find the area. [3]
3. (c) Turf is laid on the ground under the curve, between \(x=2\) and \(x=10\). Use the trapezium rule with 4 strips to estimate the area of this ground. [3]
4. (d) Use your calculator to find the exact value of \(\displaystyle\int_{2}^{10}\frac{20}{x}\,dx\). [1]
5. (e) Find the percentage error in your estimate in part (c). [2]
6. (f) The flower bed has a constant depth of 0.4 m. Find its volume. [1]