Review 7

• Area between curves for x

• Area between curves for y

• Volume –area rotated –disks for x and y

• Volume – area rotated – washers for x and y

• Volume - solid

• Area between the curves, with respect to x, for

( ( ) ( ))g x f x dxa

b

Find the area

2 1

8211

1 247

2 1

3

2

32

cos

. . . . .

. . . . . .

( co s ( ))

x

x

a

b

x x

a

b

x x dx

If it is a calculator based question, use fnint after you have written out the integral.

ln ( )

. . . . . . . .

( ln ( ) ( ))

x

x

x

a

x x dxa

2

1

4

2079

2 14

Can’t integrate ln – must use fnint

Multiple integrals (example a)

y x x x

y

a

b

c

x x x dx x x x dxb

c

a

b

3 2

3 2 3 2

5 1

2

1 903

1939

2 7093

5 1 2 2 5 1

. . . . .

. . . . . . .

. . . . . .

( ( )) ( ( )

This would be a good example to type in the equations into y1, y2 and then use the vars feature to type in y1, y2 into fnint when you get the final answer.

Store the intersections on your calculator – find one and immediately exit and store x and y – you might need both numbers

Multiple integrals (example b)x y

y x

2

12

1

1y x

y x

1

1

With respect to x, you need to rewrite as y = and have 2 integrals

( ( )) ( ( ))x x dx x x dx 1 1 1 11

0

0

8

12

Easier to write one integral with respect to y

Example b with respect to y – one integral

y x x y

y y dy

12

2

1

3

1 2 1

2 1

( )

( ( ) ( ))

Limits are with respect to y, lowest y value to the highest y value –

Store both sets of intersections for the x and the y

• Area between the curves, with respect to y, for

( ( ) ( ))g y f y dyd

c

• Area of the solid rotated around the x-axis.

( ( ))f x dxa

b

2

f(x)=x+1, bounded by x = 1, x = 4

r dx

x dx

2

1

4

2

1

4

1

( )

Area when rotated and a washer is created

( )R r dxa

b2 2

f(x)=x+1, bounded by x = 1 and x = 4 in the first quadrant, rotated around y = -1

Top minus bottom for R and top minus bottom for r

R x

r

R r dx

x dx

1 1

0 1

1 1 0 1

2 2

1

4

2 2

1

4

( )

( )

(( ( )) ( ( )) )

Area between the curves, in quadrant 1, rotated about the y-axis. Volume with

respect to y. ln x y

y

1

Rotated around y-axis

Area in quadrant Iy x x e

e dy

y

y

ln

( )0

1

2

• Area of the solid, whose base is the area between the curves, perpendicular to the x-axis, with cross sections that are squares.

• Squares area = base x base

(( ( ) ( ))g x f x dxa

b

2

2 1

8211

1 247

2 1

3

2

32 2

cos

. . . . .

. . . . . .

( co s ( ))

x

x

a

b

x x

a

b

x x dx

• Area of the solid, whose base is the area between the curves, perpendicular to the x-axis, with cross sections that are rectangles with the height twice the length.

• The length is the difference between the curves, and the height is the same set-up, but it is times’d by 2.

2 ( ( ) ( ))( ( ) ( ))g x f x g x f x dxa

b

2 1

8211

1 247

2 2 1 2 1

2 2 1

3

2

32

32

32 2

cos

. . . . .

. . . . . .

( co s ( ))( co s ( ))

( co s ( ))

x

x x

a

b

x

a

b

x x

a

b

x x x x dx

x x dx

• Area of the solid, whose base is the area between the curves, perpendicular to the x-axis, with cross sections that are semi-circles.

• The diameter is the difference between the curves, so it must be halved to get the radius for the semicircles.

2 2

2g x f x

a

b

dx( ) ( )

The diameter is the difference between the two curves, so you need to half the difference to get the radius.

Semicircles are the area cut in half