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

The graph shows part of \(y=f(x)\). The table gives some values. x: 0, 0.6, 1.2, 1.8, 2.4, 3 y: 1.8, 3.2, 3.9, 3.5, 2.6, 1.1 Estimate the area between the graph and the x-axis from x = 0 to x = 3.

Use the trapezium rule with 4 strips to estimate \(\int_{3}^{7} \sqrt{4x-11}\,dx\).

Use the trapezium rule with 5 strips to estimate \(\int_{0}^{2} \sqrt{7x+2}\,dx\).

The trapezium rule with 3 strips is used to estimate the area under \(y=5\ln x\) from x = 1 to x = 4.

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

The width of a pond is measured every 5 m along its length: 0, 12, 19, 23, 17 and 0 m. Estimate the area of the pond.

A greenhouse is 10 m wide. Its height above the ground is measured every 2.5 m across the width: 0, 2.2, 3.0, 2.2 and 0 m. The cross-section is the area between the roof 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 lake is drawn on a map. Some points on its shores are marked. Distances are in metres. Estimate the area of the lake. Top edge: (10, 84), (25, 96), (40, 91), (55, 103), (70, 95). Bottom edge: (10, 30), (25, 26), (40, 33), (55, 28), (70, 31).

The integral \(\int_{0}^{3} (x^2+1)\,dx\) is estimated with the trapezium rule using 3 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, 1.5, 3, 4.5, 6 y: 4, 6.5, k, 7.5, 3.5 The trapezium rule with 4 strips gives an estimate of 39.225 for the area under the graph. Find the value of k.