Integration: exam review Exam review: second paper

The graph shows the speed of a cyclist on a track during a journey of 22 s. It speeds up steadily from rest for the first 6 s, then travels at 9 m/s until 15 s, then slows steadily to rest. [5]

1. (a) Find the distance travelled in the first 6 s. [2]
2. (b) Find the total distance travelled. [3]

Find each indefinite integral. [7]

1. (a) \(\displaystyle\int\left(8x^{3} - \frac{15}{x^{4}} + 6\right)dx\) [3]
2. (b) \(\displaystyle\int (x+2)(3x-1)\,dx\) [2]
3. (c) \(\displaystyle\int \frac{12x^{4}-5x^{2}+9}{x^{2}}\,dx\) [2]

Show your working. [7]

1. (a) Find \(\displaystyle\int \frac{7}{x^{4}}\,dx\). [2]
2. (b) Expand \(\left(x^{3}-\dfrac{2}{x}\right)^{2}\) and hence find \(\displaystyle\int \left(x^{3}-\frac{2}{x}\right)^{2}dx\). [3]
3. (c) Given that \(\displaystyle\int g(x)\,dx=3x^{4}-\frac{8}{x^{3}}+c\), find \(g(x)\). [2]

The profile of a ski jump is modelled by a curve, with \(x\) and \(y\) in metres. The gradient of the curve is \(\dfrac{dy}{dx}=12x^{2} + 3 - \frac{9}{x^{4}}\). The curve passes through the point \((1, 14)\). [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 \(x=2\). [1]

A function \(f\) has gradient function \(f'(x)=4x^{3}-6x\), and \(f(0)=4\). [5]

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

The curve \(y=x^{2}-8x+18\) is shown. The region under the curve between \(x=2\) and \(x=6\) 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 \(\displaystyle\int_{1}^{3}\left(3x + \frac{10}{x^{2}}\right)dx\). Give your answer to 3 significant figures. [2]

For the first 30 s after a gate is lifted, a sluice releases water at a rate directly proportional to the time since it was lifted. After 4 s the rate is 36 litres per second. [4]

1. (a) Find an expression for the rate \(r\) litres per second, at time \(t\) seconds. [2]
2. (b) Find the amount of water released in the first 30 seconds. [2]

The rate at which the value of a rare coin changes is modelled by \(y=2e^{x}\) euros per year, where \(x\) is the time in years. The total change from \(x=1\) to \(x=3\) is \(\displaystyle\int_{1}^{3} 2e^{x}\,dx\). Use the trapezium rule with 4 strips to estimate this total. [7]

1. (a) Write down the values of \(y=2e^{x}\) at \(x=1.5\), \(x=2\) and \(x=2.5\), 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 reservoir. The top and bottom edges of the reservoir are marked at equal intervals along the \(x\)-axis. Points on the top edge are (0, 70), (12, 92), (24, 85), (36, 96), (48, 64). Points on the bottom edge are (0, 20), (12, 15), (24, 22), (36, 11), (48, 18). 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 \(x=48\). [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 reservoir. Give your answer to 3 significant figures. [1]

Fuel is pumped into a storage drum at a rate of \(r\) litres per minute, where \(r=kt+1.5\) 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 \(t=6\), \(r=6.3\). At \(t=0\) the drum already holds 30 litres. [12]

1. (a) Find the value of \(k\). [2]
2. (b) Find the amount in the drum when \(t=10\). [3]
3. (c) Find the value of \(t\) when the drum holds 70 litres. [3]
4. (d) The shaded region on the graph of \(r\) is the area between \(t=3\) and \(t=7\). Write down what this area represents, and find its value. [3]
5. (e) Give one criticism of this model. [1]

A train pulls out of a depot. The speed, \(v\) m/s, is modelled by \(v=4\sqrt{t}\) for \(0\le t\le 36\), where \(t\) is the time in seconds. Values of \(v\) are given to 4 significant figures. Time \(t\) (s): 0, 9, 18, 27, 36. Speed \(v\) (m/s): 0, 12, 16.97, 20.78, 24. [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 36 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 \(\displaystyle\int_0^{36} 4\sqrt{t}\,dt\). 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 train has travelled 100 m from the start. Give your answer to 3 significant figures. [2]

A reservoir is dug in a park. Its boundary is part of the curve \(y=12x-x^{2}\) and part of the line \(y=4x\), with \(x\) and \(y\) in metres. The reservoir 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 0.8 m. Find its volume. [1]

A vegetable plot is the region between the curve \(y=\dfrac{24}{x}\) and the line \(y=13-x\), 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 vegetable plot, and use your calculator to find the area. [3]
3. (c) Gravel is laid on the ground under the curve, between \(x=3\) and \(x=11\). 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 \(\displaystyle\int_{3}^{11}\frac{24}{x}\,dx\). [1]
5. (e) Find the percentage error in your estimate in part (c). [2]
6. (f) The vegetable plot has a constant depth of 0.5 m. Find its volume. [1]