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SQA Higher Maths Definite integrals

Definite integrals

This page covers the following topics:

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

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Evaluate the following integral: βˆ«β·β‚…24xΒ²dx.

SQA Higher Maths Definite integrals Evaluate the following integral: βˆ«β·β‚…24xΒ²dx.
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2

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The speed of an object is modelled by the equation 68tΒ³ + 10t, where t is time in seconds. Find the distance travelled by the object between the third and fifth second of its motion.

SQA Higher Maths Definite integrals The speed of an object is modelled by the equation 68tΒ³ + 10t, where t is time in seconds. Find the distance travelled by the object between the third and fifth second of its motion.
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3

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What is the area of the curve y = 8xΒ³ + 2 between the points x = 0 and x = 2?

SQA Higher Maths Definite integrals What is the area of the curve y = 8xΒ³ + 2 between the points x = 0 and x = 2?
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4

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The cross section of a tent is given by the graph y = βˆ’xΒ² + 10x. Find the area of the cross section between x = 3 and x = 5.

SQA Higher Maths Definite integrals The cross section of a tent is given by the graph y = βˆ’xΒ² + 10x. Find the area of the cross section between x = 3 and x = 5.
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5

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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.

SQA Higher Maths Definite integrals 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.
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6

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Find the area between y = xΒ³ + 3 and y = x + 1 between the points x = 1 and x = 2.

SQA Higher Maths Definite integrals Find the area between y = xΒ³ + 3 and y = x + 1 between the points x = 1 and x = 2.
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7

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Calculate the shaded area of the given graph.

SQA Higher Maths Definite integrals Calculate the shaded area of the given graph.
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8

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Calculate the shaded area of the given graph.

SQA Higher Maths Definite integrals Calculate the shaded area of the given graph.
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9

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State the expression for dx on the given graph.

SQA Higher Maths Definite integrals State the expression for dx on the given graph.
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10

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State the Reimann sum for the definite integral βˆ«β΄β‚5xdx.

SQA Higher Maths Definite integrals State the Reimann sum for the definite integral βˆ«β΄β‚5xdx.
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