The Definite Integral - Calculus 1 - Lecture Slides | MATH 161, Assignments of Calculus

Download The Definite Integral - Calculus 1 - Lecture Slides | MATH 161 and more Assignments Calculus in PDF only on Docsity!

The Definite Integral

MATH 161 Calculus I

J. Robert Buchanan

Department of Mathematics

Summer 2008

Background

We have seen that if f (x) ≥ 0 and continuous on the interval [a, b] the exact area under the graph of f (x), above the x-axis, and between x = a and x = b is

A = lim n→∞

n= 1

f (ci )∆x.

Remark: the limit makes sense even when f (x) < 0, but the value of the limit cannot be strictly interpreted as area.

Definite Integral

Definition For any function f defined on [a, b], the definite integral of f from a to b is ∫ (^) b

a

f (x) dx = (^) nlim→∞

n= 1

f (ci )∆x,

whenever the limit exists and is the same for any choice of evaluation points c 1 , c 2 ,... , cn. When the limit exists we say that f is integrable on [a, b].

Notation

∫ (^) b

a

f (x) dx

a: is called the lower limit of integration. b: is called the upper limit of integration. f (x): is called the integrand. dx: denotes the variable of integration.

Example

Example Use midpoint evaluation to approximate the definite integral ∫ (^3)

0

x^2 + 1 dx

n An 10 5. 6491 50 5. 6525 100 5. 6526

Example

Example Use the limit of a Riemann sum to compute the following definite integral exactly. ∫ (^3)

0

(x^2 + 1 ) dx

Illustration

a b c x

y

A 1

A 2

∫ (^) c

a

f (x) dx = A 1 − A 2

Example

Example Express as a definite integral the total area between y = 6 x − x^2 and the x-axis on the interval [ 0 , 6 ].

Application

Example An object moving along a straight line has velocity function v (t) = cos t. If the object starts at position 0, (^1) determine the total distance the object travels between t = 0 and t = 3 π/2, and (^2) determine the net distance the object travels between t = 0 and t = 3 π/2.

Properties of the Definite Integral (1 of 2)

Theorem If f is continuous on [a, b] then f is integrable on [a, b].

Remark: in fact f remains integrable even when it has a finite number of jump or removable discontinuities.

Theorem If f and g are integrable on [a, b], then (^1) for any constants c and d, ∫ (^) b

a

(cf (x) + dg(x)) dx = c

∫ (^) b

a

f (x) dx + d

∫ (^) b

a

g(x) dx,

(^2) for any c in interval [a, b], ∫ (^) b

a

f (x) dx =

∫ (^) c

a

f (x) dx +

∫ (^) b

c

f (x) dx.

Properties of the Definite Integral (2 of 2)

The following properties also hold for definite integrals: ∫ (^) b

a

f (x) dx = −

∫ (^) a

b

f (x) dx ∫ (^) a

a

f (x) dx = 0 if f (a) is defined.

Example

Example

Suppose f (x) =

−√x if x < 0 4 − x^2 if 0 ≤ x ≤ 2 and evaluate ∫ (^2)

− 2

f (x) dx.

• 2 - 1 1 2 x

y

Average Value of a Function

Suppose f is continuous on [a, b] and {x 0 , x 1 ,... , xn} is a regular partition of [a, b], then the average of {f (x 1 ), f (x 2 ),... , f (xn)} is

1 n

∑^ n

i= 1

f (xi ) =

n

∑^ n

i= 1

f (xi )

b − a n

n b − a

b − a

∑^ n

i= 1

f (xi )∆x

n^ lim→∞

n

∑^ n

i= 1

f (xi ) = (^) nlim→∞

b − a

∑^ n

i= 1

f (xi )∆x

favg =

b − a

∫ (^) b

a

f (x) dx.

Example

Example Estimate the average value of cos x on the interval [ 0 , π/ 2 ].

0.5 1.0 1. x

y