MATH 12002 - CALCULUS I 4.2: Properties of Definite Integrals - - PowerPoint PPT Presentation

MATH 12002 - CALCULUS I §4.2: Properties of Definite Integrals

Professor Donald L. White

Department of Mathematical Sciences Kent State University

D.L. White (Kent State University) 1 / 8

Algebraic Properties

The first property of definite integrals is really more of a convention. (It will be compatible with the Fundamental Theorem of Calculus.)

1

a

b

f (x) dx = − b

a

f (x) dx; that is, interchanging the limits of integration negates the integral. The next two properties follow either from the definition of the integral

• r from the area interpretation of the integral.

2

a

a

f (x) dx = 0.

3

b

a

c dx = c(b − a) for any constant c.

D.L. White (Kent State University) 2 / 8

Algebraic Properties

Since the definite integral is a limit of sums, it seems reasonable that it should “respect” sums and constant multiples as limits do.

4

b

a

[f (x) + g(x)] dx = b

a

f (x) dx + b

a

g(x) dx.

5

b

a

cf (x) dx = c b

a

f (x) dx for any constant c. EXAMPLE: If we know that 12

6 x2 dx = 504 and

12

6 x dx = 54, then

12

6

(3x2 + 2x + 5) dx = 12

6

3x2 dx + 12

6

2x dx + 12

6

5 dx = 3 12

6

x2 dx + 2 12

6

x dx + 12

6

5 dx = 3(504) + 2(54) + 5(12 − 6) = 1650.

D.L. White (Kent State University) 3 / 8

Algebraic Properties

Both properties (4) and (5) follow from the definition of the integral. For property (4), we have b

a

[f (x) + g(x)] dx = lim

n→∞ n

• i=1

[f (x∗

i ) + g(x∗ i )]∆x

= lim

n→∞

n

• i=1

f (x∗

i )∆x + n

• i=1

g(x∗

i )∆x

• =

lim

n→∞ n

• i=1

f (x∗

i )∆x + lim n→∞ n

• i=1

g(x∗

i )∆x

= b

a

f (x) dx + b

a

g(x) dx. The verification of property (5) is similar.

D.L. White (Kent State University) 4 / 8

Algebraic Properties

BE CAREFUL! This does not work for products: b

a

[f (x)g(x)] dx IS NOT EQUAL TO b

a

f (x) dx

• ·

b

a

g(x) dx

• .

Although it is true that the limit of a product is the product of the limits, the manipulations we did for the integral of a sum do not work, because

n

• i=1

[f (x∗

i ) · g(x∗ i )]∆x = n

• i=1

f (x∗

i )∆x · n

• i=1

g(x∗

i )∆x.

D.L. White (Kent State University) 5 / 8

Algebraic Properties

We can integrate over one piece of the interval at a time and then add the results to compute the integral over the whole interval.

6

b

a

f (x) dx = c

a

f (x) dx + b

c

f (x) dx. Although this is normally used in the case where a < c < b, it is valid as long as all three integrals exist. If c is not between a and b, it follows using the property that a

b

f (x) dx = − b

a

f (x) dx.

D.L. White (Kent State University) 6 / 8

Comparison Properties

The rest of the properties are used to compare and approximate integrals. They are less useful for direct calculations of integrals.

7 If f (x) 0 for a x b, then

b

a

f (x) dx 0. If f (x) 0 for a x b, then b

a

f (x) dx 0. These are useful as a rough check of calculations. Applying property (7) and the sum property to f (x) − g(x), we obtain

8 If f (x) g(x) for a x b, then

b

a

f (x) dx b

a

g(x) dx. This can be used to obtain a rough estimate for an integral if we know the integral of a larger or smaller function.

D.L. White (Kent State University) 7 / 8

Comparison Properties

We can estimate an integral if we know constant bounds on the function. Applying property (8) and the integral of a constant, we obtain

9 If m and M are numbers such that m f (x) M for a x b, then

m(b − a) b

a

f (x) dx M(b − a).

D.L. White (Kent State University) 8 / 8