Integration and the fundamental theorem of calculus – university Year 1 maths worksheet with answers to download as PDF

Find each indefinite integral, or evaluate each definite integral exactly.

1. \(\displaystyle\int(3x^2-2x+1)\,dx\) [2 marks]
2. \(\displaystyle\int_0^2(x^2+1)\,dx\) [2 marks]
3. \(\displaystyle\int_1^4\sqrt x\,dx\) [3 marks]
4. \(\displaystyle\int\left(e^{2x}+\dfrac1x\right)dx\) for \(x>0\) [2 marks]

Use a substitution to find or evaluate each integral.

1. \(\displaystyle\int\dfrac{2x}{x^2+1}\,dx\) [2 marks]
2. \(\displaystyle\int_0^1xe^{x^2}\,dx\) [3 marks]
3. \(\displaystyle\int\cos x\sin^3x\,dx\) [3 marks]

Use integration by parts, \(\int u\,v'\,dx=uv-\int u'v\,dx\), to find each integral.

1. \(\displaystyle\int xe^x\,dx\) [3 marks]
2. \(\displaystyle\int x^2\ln x\,dx\) [4 marks]
3. \(\displaystyle\int x\sin x\,dx\) [3 marks]
4. \(\displaystyle\int\ln x\,dx\) [3 marks]

Express the integrand in partial fractions and integrate.

1. \(\displaystyle\int\dfrac{dx}{x^2-1}\) for \(x>1\) [4 marks]
2. \(\displaystyle\int\dfrac{3x+5}{(x+1)(x+2)}\,dx\) for \(x>-1\) [4 marks]

Find or evaluate each integral using a trigonometric identity or substitution.

1. \(\displaystyle\int\sin^2x\,dx\) [3 marks]
2. \(\displaystyle\int_0^\pi\sin^3x\,dx\) [4 marks]
3. \(\displaystyle\int\dfrac{dx}{x^2+4}\) [3 marks]

Answer each part.

1. Find \(\dfrac d{dx}\displaystyle\int_0^{x^2}e^{t^2}dt\). [3 marks]
2. Find \(\dfrac d{dx}\displaystyle\int_x^{3}\sin(t^2)\,dt\). [2 marks]
3. Compute the left-endpoint Riemann sum with \(n=4\) for \(\displaystyle\int_0^1x^2dx\) and compare with the exact value. [4 marks]

A jogger runs along a straight track. Her velocity in m/s after \(t\) seconds is \(v(t)=t^2-4t+3\) for \(0\le t\le4\) (negative means running backwards).

1. Find her displacement from the start after 4 seconds.
2. Find when she is stationary.
3. Find the total distance she runs in the 4 seconds.
4. Explain why displacement and distance differ.

Water flows into a tank at a rate \(r(t)=12e^{-0.1t}\) litres per minute, where \(t\) is the time in minutes. The tank starts empty.

1. Write down the volume \(V(t)\) as an integral and evaluate it.
2. Find the volume after 30 minutes.
3. Find the time at which the tank holds 100 litres.
4. What is the largest volume the tank can ever hold with this inflow?

The temperature in a school hall, in \(^\circ\)C, is modelled by \(T(t)=12-6\cos\dfrac{\pi t}{12}\) for \(0\le t\le24\), where \(t\) is the number of hours after midnight.

1. Find the mean temperature over the whole day.
2. Find the mean temperature between 06:00 and 18:00 using \(\dfrac1{b-a}\int_a^bT\,dt\).
3. Why is the answer to (b) higher than (a)?
4. At what time of day is the temperature equal to the mean found in (a)?

On a map the road is the line \(y=x\) and the stream is the curve \(y=x^2\), with lengths in units of 100 m. The two meet at two points and enclose a field.

1. Find the points where the road and stream meet.
2. Find the area of the field in square units and in m\(^2\).
3. Grass seed costs 40p per m\(^2\) of field. Find the cost to the nearest £10.
4. The farmer wants fencing along the road side only. Find that length in metres.

A decorative vase is made by rotating the curve \(y=\sqrt x\) for \(0\le x\le16\) (lengths in cm) about the \(x\)-axis.

1. Show that the volume is \(V=\pi\int_0^{16}x\,dx\) and evaluate it.
2. Express the volume in millilitres (1 cm\(^3\) = 1 ml).
3. The vase is filled with sand of density 1.6 g/ml. Find the mass of sand.
4. Find the radius of the vase at the open end.

A spring in a physics laboratory exerts a force \(F=40x\) newtons when it is extended by \(x\) metres. The work done in extending it from \(a\) to \(b\) is \(\displaystyle\int_a^bF\,dx\).

1. Find the work done stretching from natural length to 0.3 m.
2. Find the work done stretching from 0.3 m to 0.5 m.
3. Compare the two results and explain why the second is larger even though the extension is only 0.2 m.
4. How far can the spring be stretched with 10 J of work from natural length?

The waiting time \(X\) (in minutes) of a passenger at a rural bus stop has density \(f(x)=\dfrac14xe^{-x/2}\) for \(x\ge0\). You are given \(\displaystyle\int_0^\infty xe^{-x/2}dx=4\).

1. Verify that \(\displaystyle\int_0^\infty f(x)\,dx=1\).
2. Find the probability of waiting less than 2 minutes: show that \(P(X<2)=1-\dfrac2e\).
3. Find the mean waiting time \(\displaystyle\int_0^\infty xf(x)\,dx\).
4. A passenger arrives. Is she more likely to wait less than 2 minutes or more than 2 minutes?

A drug is absorbed so that the rate of release into the blood is \(r(t)=50e^{-0.2t}\) mg per hour for \(t\ge0\) (in hours).

1. Find the amount released in the first 5 hours.
2. Find the total amount that will ever be released.
3. What percentage of the total is released in the first 5 hours?
4. After how many hours has 90% of the total been released?

To estimate the flow of a river, a surveyor measures the depth every 2 metres across a 12 m wide channel.

1. Use the trapezium rule with 6 strips to estimate the cross-sectional area.
2. Use Simpson's rule with 6 strips: \(\dfrac h3\left[f_0+4f_1+2f_2+4f_3+2f_4+4f_5+f_6\right]\).
3. The water flows at an average of 0.8 m/s. Find the flow in m\(^3\)/s from each method.
4. Which estimate is likely to be better, and why?

The number of fish \(P\) in a lake satisfies \(\dfrac{dP}{dt}=0.1P\left(1-\dfrac P{100}\right)\) with \(t\) in years. The time to grow from \(P_1\) to \(P_2\) is \(T=\displaystyle\int_{P_1}^{P_2}\dfrac{dP}{0.1P(1-P/100)}\).

1. Show that \(\dfrac1{0.1P(1-P/100)}=10\left(\dfrac1P+\dfrac1{100-P}\right)\).
2. Evaluate the integral to show that \(T=10\left[\ln\dfrac P{100-P}\right]_{P_1}^{P_2}\).
3. Find the time to grow from 20 fish to 50 fish.
4. Explain why the time to reach \(P_2=100\) is infinite.

Rotate the curve \(y=\dfrac1x\) for \(x\ge1\) about the \(x\)-axis to form an infinitely long horn. A painter asks whether the horn can be filled with paint and whether its surface can be painted.

1. Show that the volume is \(V=\pi\displaystyle\int_1^\infty\dfrac{dx}{x^2}\) and evaluate it.
2. The surface area is \(S=2\pi\displaystyle\int_1^\infty\dfrac1x\sqrt{1+\dfrac1{x^4}}\,dx\). Show that \(S\ge2\pi\displaystyle\int_1^\infty\dfrac{dx}x\).
3. Evaluate the integral in (b) and deduce that \(S\) is infinite.
4. Explain the apparent paradox.

In reliability theory, the lifetime of a component can be modelled by \(\displaystyle I_n=\int_0^\infty t^ne^{-t}dt\), where \(n\ge0\) is an integer.

1. Evaluate \(I_0\).
2. Use integration by parts to show that \(I_n=nI_{n-1}\) for \(n\ge1\).
3. Hence find \(I_1\), \(I_2\), \(I_3\) and \(I_4\), and guess a formula for \(I_n\).
4. Explain why \(\displaystyle\lim_{t\to\infty}t^ne^{-t}=0\) is needed in part (b).

A bowl is formed by rotating \(y=x^2\) for \(0\le y\le9\) about the \(y\)-axis (lengths in cm). It is filled from a tap at 5 cm\(^3\) per second.

1. Show that the volume of water of depth \(h\) is \(V=\dfrac\pi2h^2\).
2. Find the capacity of the bowl in ml.
3. How long does it take to fill the bowl?
4. Find the depth when the bowl is half full, and the rate at which the depth is rising at that moment.

A continuous function \(f\) satisfies \(f(x)=1+\displaystyle\int_0^xf(t)\,dt\) for all \(x\ge0\).

1. Use the fundamental theorem to show that \(f'(x)=f(x)\), and state \(f(0)\).
2. Solve the differential equation and find \(f\).
3. Check your answer by substituting back into the integral equation.
4. Starting with \(f_0(x)=1\) and \(f_{n+1}(x)=1+\int_0^xf_n(t)dt\), compute \(f_1\), \(f_2\), \(f_3\). What do they tend to?