1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Natalija [7]
2 years ago
15

4. Find the volume of the given solid bounded by the elliptic paraboloid z = 4 - x^2 - 4y^2, the cylinder x^2 + y^2 = 1 and the

plane z = 0.
5. Sketch the region of integration and change the order of integration.
Mathematics
1 answer:
Ilya [14]2 years ago
8 0

Answer:

2.5π units^3

Step-by-step explanation:

Solution:-

- We will evaluate the solid formed by a function defined as an elliptical paraboloid as follows:-

                                  z = 4 - x^2 -4y^2

- To sketch the elliptical paraboloid we need to know the two things first is the intersection point on the z-axis and the orientation of the paraboloid ( upward / downward cup ).

- To determine the intersection point on the z-axis. We will substitute the following x = y = 0 into the given function. We get:

                                 z = 4 - 0 -4*0 = 4

- The intersection point of surface is z = 4. To determine the orientation of the paraboloid we see the linear term in the equation. The independent coordinates ( x^2 and y^2 ) are non-linear while ( z ) is linear. Hence, the paraboloid is directed along the z-axis.

- To determine the cup upward or downwards we will look at the signs of both non-linear terms ( x^2 and y^2 ). Both non-linear terms are accompanied by the negative sign ( - ). Hence, the surface is cup downwards. The sketch is shown in the attachment.

- The boundary conditions are expressed in the form of a cylinder and a plane expressed as:

                                x^2 + y^2 = 1\\\\z = 4

- To cylinder is basically an extension of the circle that lies in the ( x - y ) plane out to the missing coordinate direction. Hence, the circle ( x^2 + y^2 = 1 ) of radius = 1 unit is extended along the z - axis ( coordinate missing in the equation ).

- The cylinder bounds the paraboloid in the x-y plane and the plane z = 0 and the intersection coordinate z = 4 of the paraboloid bounds the required solid in the z-direction. ( See the complete sketch in the attachment )

- To determine the volume of solid defined by the elliptical paraboloid bounded by a cylinder and plane we will employ the use of tripple integrals.

- We will first integrate the solid in 3-dimension along the z-direction. With limits: ( z = 0 , z = 4 - x^2 -4y^2 ). Then we will integrate the projection of the solid on the x-y plane bounded by a circle ( cylinder ) along the y-direction. With limits: ( y = - \sqrt{1 - x^2} , y =  \sqrt{1 - x^2} ). Finally evaluate along the x-direction represented by a 1-dimensional line with end points ( -1 , 1 ).

- We set up our integral as follows:

                            V_s = \int\int\int {} \, dz.dy.dx

- Integrate with respect to ( dz ) with limits: ( z = 0 , z = 4 - x^2 -4y^2 ):

                           V_s = \int\int [ {4 - x^2 - 4y^2} ] \, dy.dx

- Integrate with respect to ( dy ) with limits: ( y = - \sqrt{1 - x^2} , y =  \sqrt{1 - x^2} )

                        V_s = \int [ {4y - x^2.y - \frac{4}{3} y^3} ] \, | .dx\\\\V_s = \int [ {8\sqrt{( 1 - x^2 )}  - 2x^2*\sqrt{( 1 - x^2 )} - \frac{8}{3} ( 1 - x^2 )^\frac{3}{2} } ] . dx

- Integrate with respect to ( dx ) with limits: ( -1 , 1 )

                       V_s =  [ 4. ( arcsin ( x ) + x\sqrt{1 - x^2} ) - \frac{arcsin ( x ) - 2x ( 1 -x^2 )^\frac{3}{2} + x\sqrt{1 - x^2}  }{2}  - \frac{ 3*arcsin ( x ) + 2x ( 1 -x^2 )^\frac{3}{2} + 3x\sqrt{1 - x^2}  }{3} ] | \limits^1_-_1\\\\V_s =  [ \frac{5}{2} *arcsin ( x ) + \frac{5}{3}*x ( 1 -x^2 )^\frac{3}{2} + \frac{5}{2} *x\sqrt{1 - x^2} ) ] | \limits^1_-_1\\\\V_s =  [ \frac{5\pi }{2}  + 0 + 0 ] \\\\V_s = \frac{5\pi }{2}

Answer: The volume of the solid bounded by the curves is ( 5π/2 ) units^3.

Download docx
You might be interested in
Find the surface area of the figure?
motikmotik

Step-by-step explanation:

the surface area consists of 6 individual areas :

front and back (both are equal)

left and right (both are equal)

top and bottom

front and back are trapezoids.

the area of a trapezoid is

(base 1 + base 2)/2 × h

the bases are the 2 parallel sides.

in our case the area of one trapezoid is

(5 + 19)/2 × 12.1 = 145.2 ft²

front and back together then are 290.4 ft².

left and right are 8×14 rectangles:

8×14 = 112 ft²

together they are 224 ft².

the bottom is a 5×8 rectangle :

5×8 = 40 ft²

the top is a 19×8 rectangle :

19×8 = 152 ft²

so, the total surface area is

290.4 + 224 + 40 + 152 = 706.4 ft²

4 0
2 years ago
The data below represent the ages of people working at a store.
Slav-nsk [51]

Answer:

The first step is sort the values on increasing order and we got:

18, 18, 19, 19, 19, 20, 22, 23. 25,  26, 31

And in order to find the first quartile we can separate the first 5 lowest values from the largest 5 values. And for this case we focus on these values:

18, 18, 19, 19, 19

And if we find the median of the last 5 values we can find the first quartile and we got:

Q_1 = 19

The best answer would be:

19

Step-by-step explanation:

For this problem we have the following dataset:

18, 26, 19, 25, 19, 23, 31, 22, 19, 18, 20

The first step is sort the values on increasing order and we got:

18, 18, 19, 19, 19, 20, 22, 23. 25,  26, 31

And in order to find the first quartile we can separate the first 5 lowest values from the largest 5 values. And for this case we focus on these values:

18, 18, 19, 19, 19

And if we find the median of the last 5 values we can find the first quartile and we got:

Q_1 = 19

The best answer would be:

19

4 0
3 years ago
Emma went to a salon to get her hair cut. The price of the haircut is $65. She wants to leave a 20% tip. What is the total amoun
Rudiy27

Answer:

She should leave a total of $78.

Step-by-step explanation:

To find this, we first need to find the tip amount. We can do this by multiplying the total by the tip percentage.

$65 * 20% = $13

Now that we have that, we need to add it to the cost.

$65 + $13 = $78

8 0
2 years ago
Find the slope of the line passing through the points (-4, -4) and (2, 5) .
ahrayia [7]

Answer:

slope=9/8

Step-by-step explanation:

hope this help

7 0
2 years ago
Read 2 more answers
What is b’c the length of bc after dilation? What is the measure of
Anastasy [175]

Answer:

C

Step-by-step explanation:

the angle will always stay the same, it won't change

but if the dialation is 1/2, then that means that the sides will decrease in size compared to the original length,

(just divide by 2) on each side length to get the answer

4 0
3 years ago
Other questions:
  • Question 4
    15·1 answer
  • Question 7 (5 points)
    15·2 answers
  • On a coordinate plane, an absolute value graph starts at (0, 0) and goes up and to the left through (negative 4, 2). The functio
    5·2 answers
  • A line is perpendicular to<br> y = - 1/4x-9<br> What is the slope of this<br> perpendicular line?
    7·1 answer
  • Please Help Me!!!!!!!​
    7·1 answer
  • What is the slope of the line that contains the points (13,2) and (3,-2)
    13·1 answer
  • Kenya bought 4 boxes of
    13·1 answer
  • What divided by five is seven
    6·2 answers
  • Need help with science
    10·1 answer
  • The base of a garden box has an area of 27,404 square centimeters. It's height is 18 centimeters. What is the volume of the gard
    12·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!