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Definite integrals for SQA Higher Maths

Definite integrals

This page covers the following topics:

1. Definite integrals
2. Integrating area under a curve
3. Integrating area between two curves
4. Integration limits

A definite integral is one with limits of integration. To evaluate a definite integral, integrate the expression and write it in square brackets, placing the limits after the closing bracket. Substitute the limits into the expression and subtract the one from the other to calculate the integral.

Definite integrals

Definite integrals can be used to find the area under a curve by setting the limits of integration to be the two points between which the area is wanted. A negative area indicates that that section of the graph lies under the x axis.

Integrating area under a curve

To find the area between two curves, definite integrals are used. The lower function is subtracted from the upper function and the definite integral is evaluated between the two points required.

Integrating area between two curves

Evaluating the definite integral between two points gives the exact value of the area under the curve between those points. The area under the curve can be approximated by dividing the area up into rectangles and summing their areas up. The more rectangles the area is divided into, the more accurate the approximation will be. The exact value of the definite integral and thus the area under the curve is the limit of the sum of the areas of the rectangles of equal width as the number of rectangles tends to infinity. This is called the Reimann sum.

Integration limits

1

Use the formula of the area of a triangle and the given graph to prove that the area under the curve is given by evaluating the integral between two points.

Using the formula for the area of a triangle, area = (3 ร— 21)/2 = 31.5.
Evaluating the definite integral, โˆซยณโ‚€7xdx = [7xยฒ/2]ยณโ‚€ = 7(3)ยฒ/2 โˆ’ 7(0)ยฒ/2 = 31.5.
The two areas are equal, therefore evaluating the definite integral between two points gives the area under the curve between those points.

Use the formula of the area of a triangle and the given graph to prove that the area under the curve is given by evaluating the integral between two points.

2

Find the definite integral of 12cos(3x) with limits of integration ฯ€/6 and 2ฯ€.

โˆซ12cos(3x)dx = 4sin(3x) + c
To find the definite integral, 4sin(6ฯ€) โˆ’ 4sin(ฯ€/2) = โˆ’4.

โˆ’4

Find the definite integral of 12cos(3x) with limits of integration ฯ€/6 and 2ฯ€.

3

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.

4

Explain what is meant by a Reimann sum.

The area under the curve can be approximated by dividing the area up into rectangles and summing their areas up. The exact value of the area under the curve is the limit of the sum of the areas of the rectangles of equal width as the number of rectangles tends to infinity, which is called the Reimann sum.

Explain what is meant by a Reimann sum.

5

Find the area under the curve given by y = 15 โˆ’ xยฒ between x = 12 and x = 10.

To find the area of the curve between those points, the definite integral between them must be found.
โˆซยนยฒโ‚โ‚€15 โˆ’ xยฒdx = [15x โˆ’ xยณ/3]ยนยฒโ‚โ‚€ = 15(12) โˆ’ (12)ยณ/3 โˆ’ (15(10) โˆ’ (10)ยณ/3) = โˆ’638/3.

Find the area under the curve given by y = 15 โˆ’ xยฒ between x = 12 and x = 10.

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