The trapezium rule Problem set A

Use the trapezium rule with 4 strips to estimate \(\int_{0}^{4} \frac{1}{x+3}\,dx\).

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

Use the trapezium rule with 5 strips to estimate \(\int_{0}^{1} 8e^x\,dx\).

Use the trapezium rule with 4 strips to estimate \(\int_{3}^{5} \ln x\,dx\).

Use the trapezium rule with 4 strips to estimate \(\int_{2}^{4} \ln(4x)\,dx\).

The table shows the rate R (litres per second) at which water flows from a tap, measured every 3 seconds. The area under the graph is the volume of water. t: 0, 3, 6, 9, 12 R: 0, 1.8, 2.4, 2.1, 1.1 Estimate the area under the graph.

A sprinter's speed v (m/s) is measured every 0.5 seconds at the start of a race. t: 0, 0.5, 1, 1.5, 2, 2.5 v: 0, 2.9, 4.6, 5.8, 6.7, 7.3 Estimate the distance run in the first 2.5 seconds.

A pool is filled at the rate \(r=\ln(3t)\) hundred litres per minute, for \(1\le t\le 4\), where t is in minutes. Use the trapezium rule with 6 strips to estimate the volume added, in hundreds of litres, between t = 1 and t = 4.

The integral \(\int_{1}^{6} \frac{1}{x}\,dx\) 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 \(\int_{0}^{2} 3e^x\,dx\) is to be estimated.

1. (a) Use the trapezium rule with 2 strips.
2. (b) Use the trapezium rule with 4 strips.
3. (c) Find the value of the integral with your calculator. By how much does each estimate differ from it?