The trapezium rule Problem set B

Use the trapezium rule with 6 strips to estimate \(\int_{0}^{3} \frac{1}{x+6}\,dx\).

Use the trapezium rule with 6 strips to estimate \(\int_{2}^{5} \frac{1}{x}\,dx\).

Use the trapezium rule with 5 strips to estimate \(\int_{0}^{1.5} 5e^x\,dx\).

Use the trapezium rule with 6 strips to estimate \(\int_{4}^{7} \ln x\,dx\).

Use the trapezium rule with 4 strips to estimate \(\int_{1}^{3} \ln(6x)\,dx\).

The table shows the speed v (m/s) of a cyclist, measured every 10 seconds. The area under the graph is the distance travelled. t: 10, 20, 30, 40, 50, 60 v: 3.2, 5.1, 6.4, 6.9, 6, 4.4 Estimate the area under the graph.

Water flows into a tank at a rate R (litres per minute), measured every 5 minutes. t: 0, 5, 10, 15, 20 R: 12, 9.4, 7.9, 6.1, 5.2 Estimate the volume of water that flows in during the 20 minutes.

Oil leaks from a pipe at the rate \(r=\ln(5t)\) litres per hour, for \(1\le t\le 3.5\), where t is in hours. Use the trapezium rule with 5 strips to estimate the volume that leaks between t = 1 and t = 3.5.

The integral \(\int_{2}^{7} \frac{1}{x}\,dx\) is to be estimated.

1. (a) Use the trapezium rule with 5 strips.
2. (b) Use the trapezium rule with 10 strips.
3. (c) Use your calculator to find the value of the integral. Which estimate is closer to it, and why?

The integral \(\int_{0}^{3} 2e^x\,dx\) is to be estimated.

1. (a) Use the trapezium rule with 3 strips.
2. (b) Use the trapezium rule with 6 strips.
3. (c) Find the value of the integral with your calculator. By how much does each estimate differ from it?