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Numerical integration for SQA Higher Maths

Numerical integration

This page covers the following topics:

1. Calculating area under a curve

The area bounded between two curves can be found by subtracting the lower function by the upper one and integrating between the limits.

Calculating area under a curve

1

Calculate the shaded area of the given graph.

The area is given by the integral โˆซโฐโ‚‹โ‚xยณ + xยฒ + 5 โˆ’ (xยฒ/2 + 4)dx = โˆซโฐโ‚‹โ‚xยณ + xยฒ/2 + 1dx = [xโด/4 + xยณ/6 + x]โฐโ‚‹โ‚ = 0โด/4 + 0ยณ/6 + 0 โˆ’ ((โˆ’1)โด/4 + (โˆ’1)ยณ/6 โˆ’ 1) = 11/12.

Calculate the shaded area of the given graph.

2

Find the area between y = xยณ + 3 and y = x + 1 between the points x = 1 and x = 2.

The area is given by the integral โˆซยฒโ‚xยณ + 3 โˆ’ (x + 1)dx = โˆซยฒโ‚xยณ โˆ’ x + 4dx = [xโด/4 โˆ’ xยฒ/2 + 4x]ยฒโ‚ = 2โด/4 โˆ’ 2ยฒ/2 + 4(2) โˆ’ (1โด/4 โˆ’ 1ยฒ/2 + 4(1)) = 25/4.

Find the area between y = xยณ + 3 and y = x + 1 between the points x = 1 and x = 2.

3

Use definite integrals to find the area between y = 2x + 10 and y = 2x + 4 between the points x = 5 and x = 2.

The area is given by the integral โˆซโตโ‚‚2x + 10 โˆ’ (2x + 4)dx = โˆซโตโ‚‚6dx = [6x]โตโ‚‚ = 6(5) โˆ’ 6(2) = 18.

Use definite integrals to find the area between y = 2x + 10 and y = 2x + 4 between the points x = 5 and x = 2.

4

Evaluate the area between the curves y = โˆ’x + 9 and y = xยฒ + 3 between the points x = 0 and x = 2.

The area is given by the integral โˆซยฒโ‚€โˆ’x + 9 โˆ’ (xยฒ + 3)dx = โˆซยฒโ‚€โˆ’x + 6 โˆ’ xยฒdx = [โˆ’xยฒ/2 + 6x โˆ’ xยณ/3]ยฒโ‚€ = โˆ’2ยฒ/2 + 6(2) โˆ’ 2ยณ/3 โˆ’ (โˆ’0ยฒ/2 + 6(0) โˆ’ 0ยณ/3) = 22/3.

Evaluate the area between the curves y = โˆ’x + 9 and y = xยฒ + 3 between the points x = 0 and x = 2.

5

Find the area between y = โˆ’x and y = xยฒ โˆ’ 8 between the points x = 0 and x = 2.

The area is given by the integral โˆซยฒโ‚€xยฒ โˆ’ 8 โˆ’ โˆ’xdx = โˆซยฒโ‚€xยฒ + x โˆ’ 8dx = [xยณ/3 + xยฒ/2 โˆ’ 8x]ยฒโ‚€ = 2ยณ/3 + 2ยฒ/2 โˆ’ 8(2) โˆ’ (0ยณ/3 + 0ยฒ/2 โˆ’ 8(0)) = โˆ’34/3.

Find the area between y = โˆ’x and y = xยฒ โˆ’ 8 between the points x = 0 and x = 2.

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