Integration – Year 12 A level maths worksheet with answers to download as PDF

Find each integral. Remember the constant of integration.

1. \(\displaystyle\int(6x^2-4x+5)\,dx\) [2 marks]
2. \(\displaystyle\int\left(3\sqrt x-\dfrac2{x^2}\right)dx\) [3 marks]
3. \(\displaystyle\int(2x+3)(x-1)\,dx\) [3 marks]
4. \(\displaystyle\int\dfrac{4x^3-x}x\,dx\) [3 marks]

Use the given point to find the constant of integration.

1. The curve \(y=f(x)\) has \(\dfrac{dy}{dx}=6x-4\) and passes through \((1,3)\). Find \(y\). [4 marks]
2. \(f'(x)=3x^2-2x\) and \(f(2)=5\). Find \(f(x)\). [4 marks]
3. Find the equation of the curve with gradient \(\dfrac{dy}{dx}=\dfrac4{\sqrt x}\) passing through \((4,10)\). [4 marks]

Evaluate each definite integral.

1. \(\displaystyle\int_1^3(2x+1)\,dx\) [3 marks]
2. \(\displaystyle\int_0^4\sqrt x\,dx\) [3 marks]
3. \(\displaystyle\int_1^2\left(x^2-\dfrac1{x^2}\right)dx\) [4 marks]

Find the area of each region.

1. The region between \(y=x^2+2\), the \(x\)-axis and the lines \(x=1\) and \(x=3\). [4 marks]
2. The region enclosed by \(y=4x-x^2\) and the \(x\)-axis. [5 marks]

The curve \(y=x^2-4x+3\) crosses the \(x\)-axis at \(x=1\) and \(x=3\).

1. Find \(\displaystyle\int_1^3(x^2-4x+3)\,dx\) and explain the sign. [3 marks]
2. Find the area enclosed between the curve and the \(x\)-axis. [1 mark]
3. Find the total area between the curve, the \(x\)-axis, \(x=0\) and \(x=3\). [4 marks]

The curve \(y=x^2\) and the line \(y=2x\) meet at two points.

1. Find the coordinates of the points of intersection. [2 marks]
2. Find the area of the region enclosed between the curve and the line. [4 marks]

Water flows into a tank at a rate of \(\dfrac{dV}{dt}=20-2t\) litres per minute for \(0\le t\le10\), where \(t\) is in minutes. The tank initially holds 5 litres.

1. Show that \(V=5+20t-t^2\). [2 marks]
2. How much water is in the tank after 10 minutes? [2 marks]
3. How much water enters between \(t=2\) and \(t=6\)? [3 marks]
4. The tank holds 120 litres when full. Will it overflow during these 10 minutes? [2 marks]

A particle moves on a straight line with velocity \(v=3t^2-12t+9\) m/s for \(0\le t\le4\). Its displacement from the origin is \(s\) metres, with \(s=0\) when \(t=0\).

1. Show that \(s=t^3-6t^2+9t\). [2 marks]
2. Find the times when the particle is at rest. [2 marks]
3. Find the displacement at \(t=1\), \(t=3\) and \(t=4\). [3 marks]
4. Find the total distance travelled in the first 4 seconds. [3 marks]

A garden designer plots a flower bed on a plan with units in metres. The bed lies between the \(x\)-axis and the curve \(y=6x-x^2\) for \(0\le x\le6\).

1. Find the area of the bed. [3 marks]
2. The bed is to be filled to a depth of 0.3 m with soil. Find the volume of soil. [1 mark]
3. Soil costs £42 per cubic metre. Find the cost. [1 mark]
4. Find the greatest width of the bed (in the \(y\)-direction). [2 marks]

The marginal cost (cost of one extra item) of producing \(x\) items is \(\dfrac{dC}{dx}=0.06x+4\) pounds. There are fixed costs of £500.

1. Show that \(C=0.03x^2+4x+500\). [2 marks]
2. Find the cost of producing 100 items. [1 mark]
3. Find the extra cost of increasing production from 100 to 150 items. [3 marks]
4. Find the average cost per item when 150 items are made. [2 marks]

An arched window has a straight bottom edge on the \(x\)-axis from \((0,0)\) to \((8,0)\) and a top edge following \(y=4x-\tfrac12x^2\) (units: feet).

1. Find the height of the window at its centre. [2 marks]
2. Find the area of the window. [4 marks]
3. Glass costs £85 per square foot. Find the cost to the nearest £10. [2 marks]
4. Find the area of the rectangle enclosing the window and the fraction filled by the window. [3 marks]

The rate of growth of a town\'s population is \(\dfrac{dP}{dt}=100+12t\) people per year, where \(t\) is the number of years since 2020. The population in 2020 was 5 000.

1. Show that \(P=5000+100t+6t^2\). [2 marks]
2. Find the population in 2030. [1 mark]
3. How many people are added between 2025 and 2030? [3 marks]
4. In which year does the population first exceed 7 000? [4 marks]

On a map (units of 100 m) a road follows \(y=x\) and a river follows \(y=4x-x^2\). They meet at the origin and at another point.

1. Find the other point of intersection. [3 marks]
2. Find the area between the road and the river. [4 marks]
3. Convert to hectares (1 unit\(^2\) = 10 000 m\(^2\) = 1 hectare). [1 mark]
4. Land costs £28 000 per hectare. Find the price. [1 mark]

The temperature in a greenhouse (°C) is modelled by \(T=-0.5t^2+6t+8\) for \(0\le t\le12\), where \(t\) is the number of hours after 6 am.

1. Find the maximum temperature and when it occurs. [3 marks]
2. The mean value of \(T\) over the interval is \(\dfrac1{12}\int_0^{12}T\,dt\). Find it. [4 marks]
3. At what times is the temperature exactly equal to the mean value? [4 marks]

The curve \(y=x(4-x)\) is shown by the table of values.

1. Check two entries of the table. [2 marks]
2. Estimate the area under the curve between \(x=0\) and \(x=4\) by joining the points with straight lines (trapezia of width 1). [3 marks]
3. Find the exact area by integration. [3 marks]
4. Find the percentage error of the estimate, and explain why it is an underestimate. [3 marks]

A curve has gradient \(\dfrac{dy}{dx}=ax+b\), where \(a\) and \(b\) are constants. It has a stationary point at \(x=2\) and passes through \((0,1)\) and \((2,-3)\).

1. Show that \(y=\tfrac12ax^2+bx+1\). [2 marks]
2. Find \(a\) and \(b\) and hence \(y\). [5 marks]
3. The curve crosses the \(x\)-axis at \(x=2\pm\sqrt3\). Show that the area enclosed with the \(x\)-axis is \(4\sqrt3\). [5 marks]

The curves \(C_1\): \(y=6-x^2\) and \(C_2\): \(y=x^2-2\) enclose a region.

1. Show that the curves meet where \(x=\pm2\). [2 marks]
2. Find the area of the enclosed region. [5 marks]
3. On a plan the units are 10 m. Convert the area to square metres. [1 mark]
4. A pond liner costs £6.50 per square metre. Find the cost to the nearest £100. [2 marks]

Find the value of the unknown in each part.

1. \(\displaystyle\int_1^k(2x+3)\,dx=18\) with \(k>1\). Give \(k\) to 2 decimal places. [5 marks]
2. \(\displaystyle\int_0^p(6-2x)\,dx=8\). Find the possible values of \(p\) and explain geometrically why there are two. [6 marks]

A cyclist\'s velocity along a straight road is \(v=t^2-6t+5\) m/s for \(0\le t\le6\). Positive means forwards.

1. Show that the cyclist is momentarily at rest at \(t=1\) and \(t=5\). [2 marks]
2. Find the displacement from the start at \(t=6\). [3 marks]
3. Find the total distance cycled in the 6 seconds. [5 marks]
4. How far from the start is the cyclist when furthest back? [2 marks]

The region \(R\) is bounded by the curve \(y=x(a-x)\) and the \(x\)-axis, where \(a>0\).

1. Show that the area of \(R\) is \(\dfrac{a^3}6\) and find \(a\) when the area is 36. [4 marks]
2. With \(a=6\), the line \(y=mx\) (\(0\lt m<6\)) cuts \(R\) into two parts. Show that the area between the curve and the line is \(\dfrac{(6-m)^3}6\). [5 marks]
3. Find \(m\) so that the line splits \(R\) into two equal areas, giving \(m\) to 3 significant figures. [4 marks]