The trapezium rule with data, and its error Problem set A

The graph shows part of \(y=f(x)\). The table gives some values. x: 1, 1.5, 2, 2.5, 3, 3.5 y: 2.4, 3.8, 4.5, 4.1, 3, 1.6 Estimate the area between the graph and the x-axis from x = 1 to x = 3.5.

Use the trapezium rule with 4 strips to estimate \(\int_{1}^{5} \sqrt{2x-1}\,dx\).

Use the trapezium rule with 5 strips to estimate \(\int_{0}^{3} \sqrt{5x+1}\,dx\).

The trapezium rule with 3 strips is used to estimate the area under \(y=\frac{12}{x+1}\) from x = 0 to x = 3.

1. (a) Find the estimate.
2. (b) Is it an overestimate or an underestimate? Give a reason.

The width of a lake is measured every 20 m along its length: 0, 45, 68, 82, 74 and 0 m. Estimate the area of the lake.

A bridge arch is 12 m wide. Its height above the ground is measured every 3 m from one end: 0, 2.8, 4.1, 3.3 and 0 m. The opening under the arch is the area between the arch and the ground.

1. (a) Estimate the cross-sectional area.
2. (b) The roof curves outwards between the measured points. Is the real area larger or smaller than your estimate? Explain.

A field is drawn on a map. Some points on its edges are marked. Distances are in metres. Estimate the area of the field. Top edge: (20, 150), (40, 172), (60, 160), (80, 181), (100, 168). Bottom edge: (20, 62), (40, 58), (60, 66), (80, 61), (100, 64).

The integral \(\int_{0}^{2} (x^3+x)\,dx\) is estimated with the trapezium rule using 4 strips.

1. (a) Find the estimate.
2. (b) Use your calculator to find the exact value of the integral.
3. (c) Find the percentage error of the estimate.

A table gives some values of \(y=f(x)\). x: 0, 2, 4, 6, 8 y: 5, 7.5, k, 6.5, 3 The trapezium rule with 4 strips gives an estimate of 54.4 for the area under the graph. Find the value of k.