Calc 4.2b

4.2b AreaRevisited

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y = -x2 +5 Find area from x=0 to 2

Width:

Heights: f( ), f( ), f( ), . . .

ith height? f( )

b a

n

Area ≈∑

2

5

4

5

10

5

6

5

8

5

This would be called a lower sum, since it is an underestimate and all the rectangles are formed under the curve. It could also be called a right-hand sum, since all the rectangles are formed by the heights at the right hand side of the rectangle.

2 0

5

2

5

Width:

Heights: f( ), f( ), f( ), . . .

ith height? f( )

b a

n

2 0

5

2

5

y = -x2 +5 Find area from x=0 to 2

Area ≈

This would be called an upper sum, since it is an overestimate and all the rectangles are formed above the curve. It could also be called a left-hand sum, since all the rectangles are formed by the heights at the left hand side of the rectangle.

0

5

2

5

4

5

6

5

8

5

Aii

1

5

xb a

n

f(mi)=Minimum height for ith interval

f(Mi)=Maximum height for ith interval

area of inscribed

rectangle

area of circumscribed

rectangle f m x f M xi i( ) ( )

Lower sum = s n f m xii

n

( ) ( )

1

Upper sum = S n f M xii

n

( ) ( )

1

Refer to p. 263

Refer to p. 263

Find the upper and lower sums for the region bounded by the graph of f(x) = x2 and the x-axis between x = 0 and x = 2

xb a

n

Since f is increasing on interval, lower sum rectangles form from the left endpoint of each interval.

Left endpoints:m a ib a

ni F

HGIKJ( )1 Right endpoints:m a i

b a

ni F

HGIKJ( )

s n f m xii

n

( ) ( )

1

S n f M xii

n

( ) ( )

1

f(x) = x2

Upper sum rectangles form from the right endpoint of each interval

Ex 4 p. 264

s nn n n n

S n( ) ( ) 8

3

4 4

3

8

3

4 4

32 2

As n increases, these two sums get closer to the same value.

limn n n

FHG

IKJ

8

3

4 4

3

8

32 limn n n

FHG

IKJ

8

3

4 4

3

8

32

Refer to p. 265

Refer to p. 265

Ex 5 p. 266 Finding area by using the limit definition

Find area under graph f(x) = x3, above the x-axis, and between x=0 and x = 1

Area = lim ( ) limn

ii

n

ni

n

f c x

1 1

xb a

n

choose rt endpoints c a ib a

ni F

HGIKJ( )

Ex 6 p. 266 Finding area by using the limit definition

Find area under graph f(x) = 4 – x2, above the x-axis, and between x=1 and x = 2

xb a

n

choose rt endpoints c a ib a

ni F

HGIKJ( )

Area = lim ( ) limn

ii

n

ni

n

f c x

1 1

Ex 7 p. 267 A region bounded by y-axis

Find the area of the region bounded by the graph of f(y) = y2, the y-axis, and 0 ≤ y ≤ 1

When f is a continuous, nonnegative function of y, you can still use same techniques.

Area = lim ( ) limn

ii

n

ni

n

f c y

1 1

yb a

n n

1 c a ib a

nini

FHG

IKJ

http://www.math.psu.edu/dlittle/java/calculus/area.html

4.2b p. 267/ 23-29 odd, 33-42 mult of 3, 49, 53, 59

A great resource for visualizing and checking answers is:

http://youtu.be/ZXKowQRwuwA for left hand sums