Find the volume of the given solid. under the surface z = 6xy and above the triangle with vertices (1, 1), (4, 1), and (1, 2)

he volume of the solid under a surface  

z

=

f

(

x

,

y

)

and above a region D is given by the formula  

∫

∫

D

f

(

x

,

y

)

d

A

.

Here  

f

(

x

,

y

)

=

6

x

y

. The inequalities that define the region D can be found by making a sketch of the triangle that lies in the  

x

y

−

plane. The bounding equations of the triangle are found using the point-slope formula as  

x

=

1

,

y

=

1

and  

y

=

−

x

3

+

7

3

.

Here is a sketch of the triangle:

Intersecting Region

The inequalities that describe D are given by the sketch as:  

1

≤

x

≤

4

and  

1

≤

y

≤

−

x

3

+

7

3

.

Therefore, volume is

V

=

∫

4

1

∫

−

x

3

+

7

3

1

6

x

y

d

y

d

x

=

∫

4

1

6

x

[

y

2

2

]

−

x

3

+

7

3

1

d

x

=

3

∫

4

1

x

[

y

2

]

−

x

3

+

7

3

1

d

x

=

3

∫

4

1

x

[

49

9

−

14

x

9

+

x

2

9

−

1

]

d

x

=

3

∫

4

1

40

x

9

−

14

x

2

9

+

x

3

9

d

x

=

3

[

40

x

2

18

−

14

x

3

27

+

x

4

36

]

4

1

=

3

[

(

640

18

−

896

27

+

256

36

)

−

(

40

18

−

14

27

+

1

36

)

]

=

23.25

.

Volume is  

23.25

.

the arc length is found by finding the circumference of the full circle (2xpixradius) and then dividing by 4 to find just one quarter of the circle (90 degrees).

answer:   35 % is the probability of a late departure or an early arrival.

step-by-step explanation:

let l represents the late departure and e represents the early departure.

then, according to the question,

p(l) = 12 %,   p(e) = 27 % and = 4 %

and, we have to find out,

but, we know that,

which is the required equation that will use to find the probability of a late departure or an early departure.

therefore,   the probability of a late departure or an early arrival =   35 %