5.2 Definite Integrals

Greg Kelly, Hanford High School, Richland, Washington

5.2Definite Integrals

WHAT YOU’LL LEARN ABOUT Riemann Sums The Definite Integral Computing Definite Integrals on a

Calculator Integrability

… and whyThe definite integral is the basis of integral

calculus, just as the derivative is the basis of

differential calculus. Slide 5-

3

QUICK REVIEW

Slide 5- 4

42

1

4

1

3 3 3

42

1

Evaluate the sum.

1.

2. 3 1

Write the sum in sigma notation.3. 2 3 4 ... 49 504. 2 4 6 8 ... 98 1005. 3(1) 3(2) ... 3(100)

6. Write the expression as a single sum in sigma notation

n

k

n

n

k

n

4

1

0

0

3

7. Find 1 if is odd.

8. Find 1 if is even.

n

n k

k

n k

k

n

n

n

QUICK REVIEW SOLUTIONS

50

4

2

50

1

1003

1

2

1

4

1

3 3 3

Evaluate the sum.

1.

2. 3 1

Write the sum in sigma notation.

3. 2 3 4 ... 49 50

4. 2 4 6 8 ... 98 100

5. 3(1) 3(2) ... 3(100)

6. Writ

30

34

2

e the expres

3

k

k

k

n

k

n

k

k

k

k

4 42

1 1

0

0

42

1sion as a single sum in sigma notation 3

7. Find 1 if is odd.

8. Find 1 if is even.

3

0

1

nn n

n k

k

n k

k

n n

n

n n

n

Slide 5- 5

When we find the area under a curve by adding rectangles, the answer is called a Rieman sum.

21 18

V t

subinterval

partition

The width of a rectangle is called a subinterval.

The entire interval is called the partition.

Subintervals do not all have to be the same size.

21 18

V t

subinterval

partition

If the partition is denoted by P, then the length of the longest subinterval is called the norm of P and is denoted by .P

As gets smaller, the approximation for the area gets better.

P

0 1

Area limn

k kP k

f c x

if P is a partition of the interval ,a b

0 1

limn

k kP k

f c x

is called the definite integral of

over .f ,a b

If we use subintervals of equal length, then the length of a

subinterval is:b axn

The definite integral is then given by:

1

limn

kn k

f c x

1

limn

kn k

f c x

Leibniz introduced a simpler notation for the definite integral:

1

limn b

k an k

f c x f x dx

Note that the very small change in x becomes dx.

b

af x dx

IntegrationSymbol

lower limit of integration

upper limit of integration

integrandvariable of integration

(dummy variable)

b

af x dx

We have the notation for integration, but we still need to learn how to evaluate the integral.

time

velocity

After 4 seconds, the object has gone 12 feet.

In section 5.1, we considered an object moving at a constant rate of 3 ft/sec.

Since rate . time = distance: 3t d

If we draw a graph of the velocity, the distance that the object travels is equal to the area under the line.

ft3 4 sec 12 ftsec

If the velocity varies:

1 12

v t

Distance:21

4s t t

(C=0 since s=0 at t=0)

After 4 seconds:1 16 44

s

8s

1Area 1 3 4 82

The distance is still equal to the area under the curve!

Notice that the area is a trapezoid.

21 18

v t What if:

We could split the area under the curve into a lot of thin trapezoids, and each trapezoid would behave like the large one in the previous example.

It seems reasonable that the distance will equal the area under the curve.

21 18

dsv tdt

3124

s t t

31 4 424

s

263

s

The area under the curve263

We can use anti-derivatives to find the area under a curve!

AREA

Slide 5- 16

Area= ( ) when ( ) 0.

( ) area above -axis area below -axis .

ba

ba

f x dx f x

f x dx x x

THE INTEGRAL OF A CONSTANT

Slide 5- 17

If ( ) , where is a constant, on the interval [ , ], then( ) ( ) b b

a a

f x c c a bf x dx cdx c b a

EXAMPLE USING NINT

Slide 5- 18

2

-1Evaluate numerically. sinx xdx

NINT( sin , , -1,2) 2.04x x x