# StudySquare

# SQA Higher Maths 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 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.

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.

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.

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The speed of an object is modelled by the equation 68t³ + 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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The cross section of a tent is given by the graph y = −x² + 10x. Find the area of the cross section between x = 3 and x = 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.

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