Do Now: p.381: #8Integrate the two parts separately:

1 22

0 12x dx x dx Shaded Area =

1 23 2

0 1

23 2x x x

1 10 2 4 23 2

56

Integrating with Respect

to ySection 7.2b

Integrating with Respect to ySometimes the boundaries of a region are more easilydescribed by functions of y than by functions of x. In suchcases, we can use horizontal rectangles:

d

c x g y

x f y d

c

x g y

x f y

d

cA f y g y dy

Returning to an example from last class…Find the area of the region R in the first quadrant that is boundedabove by and below by the x-axis and the line .y x 2y x

y x2y x

(4,2)

2 4

1

2

f y g y

y

,g y y

,f y y

0y

Solve our previousequations for x:

2x y2x y 0y

2x y

2x y

2 2

02A y y dy

103

22 3

0

22 3y yy

Returning to an example from last class…And a third way to handle this example: Using geometry formulas.

(4,2)

2 4

1

2

0y

Integrate the square rootfunction from 0 to 4, thensubtract the area of thetriangle on the right:

y x2y x

4

0

1 2 22

A xdx 103

4

3 2

0

2 23x

2

2

Quality Practice ProblemsTry #4 from p.380:

1 2 3 2

012 12 2 2A y y y y dy

43

1 3 2

012 10 2y y y dy

14 3 2

0

1033

y y y 103 13

Quality Practice ProblemsFind the area of the region enclosed by the given functions.

2x y 2x y How about the graph?

(4,2)

(1,–1)

To find the y-coordinates of theIntersection points, solve:

2 2y y 1, 2y y

2 2

12A y y dy

92

8 1 12 4 23 2 3

22 3

1

22 3y yy

Quality Practice ProblemsFind the area of the region enclosed by the given functions.

3y x 2 2x y Use your calculator to graph in [–3, 3] by [–3, 3].

(c,d)

(a,b)

31y x 2 2y x 3 2y x

We could integrate with respect to x,but that would require splitting theregion at x = a.

Instead, let’s integrate from y = b toy = d, handling the entire region atonce…

Quality Practice ProblemsFind the area of the region enclosed by the given functions.

3y x 2 2x y Use your calculator to graph in [–3, 3] by [–3, 3].

(c,d)

(a,b)

31y x 2 2y x 3 2y x

3x y

2 2x y

1 3 2NINT 2 , , 1,DA y y y

1.793003715d

Calculate b and d, storing the valuesin your calculator:

1b

Evaluate the area numerically:

4.215

Do Now: p.381: #8Returning to the “Do Now,” let’s use our new technique!!!

1

02 y y dy Shaded Area =

56

12 3 2

0

1 222 3

y y y

1 22 02 3

Quality Practice Problems23y x Find the area of the region between the curve and

the line by integrating with respect to (a) x, (b) y.1y

2 2

02 3 1A x dx

(2,–1)

Integrate with respect to x:

2 2

02 4 x dx

23

0

12 43

x x

Graph the region:

(–2,–1)

23y x

1y

82 8 03

323

Quality Practice Problems23y x Find the area of the region between the curve and

the line by integrating with respect to (a) x, (b) y.1y

(2,–1)

Graph the region:

(–2,–1) 1y

3

12 3A ydy

Integrate with respect to y:

3

3 2

1

22 33

y

3x y

162 03

323

4y xFind the area of the region in the first quadrant bounded on theleft by the y-axis, below by the line , above left by thecurve , and above right by the curve 2y x

1

01

4xA x dx

Integrate in two parts:

4

1

24x dx

x

Graph the region:

x = 1

1y x

4y x

2y x

1y x

x = 41 42 2

3 2

0 1

2 43 8 8

x xx x x

4y xFind the area of the region in the first quadrant bounded on theleft by the y-axis, below by the line , above left by thecurve , and above right by the curve 2y x

Graph the region:

x = 1

1y x

4y x

2y x

1y x

x = 4

42

1

48xx

123 2

0

23 8

xx x

113

2 1 11 8 2 43 8 8