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Eduardwww [97]
3 years ago
12

Write the integral in one variable to find the volume of the solid obtained by rotating the first-quadrant region bounded by y =

0.5x2 and y = x about the line x = 3.
Mathematics
1 answer:
Dovator [93]3 years ago
7 0

Answer:

V = π ∫₀² (y² − 8y + 6√(2y)) dy

or

V = π ∫₀² (6x − 5x² + x³) dx

Step-by-step explanation:

y₁ = 0.5x²

y₂ = x

First, find the intersections of the curves.

0.5x² = x

x² = 2x

x² − 2x = 0

x (x − 2) = 0

x = 0 or x = 2

So the points of intersection are (0, 0) and (2, 2).

When we revolve this region about the line x = 3, we get a hollow shape that looks like an upside-down funnel, or a volcano.

One option is to use washer method to find the volume, by cutting a thin horizontal slice of thickness dy, inner radius 3−x₁ = 3−√(2y), and outer radius of 3−x₂ = 3−y.

V = ∫₀² π [(3−y)² − (3−√(2y))²] dy

V = ∫₀² π (9 − 6y + y² − 9 + 6√(2y) − 2y) dy

V = π ∫₀² (y² − 8y + 6√(2y)) dy

Another option is to use shell method to find the volume, by cutting a thin vertical slice of thickness dx, radius 3−x, and height y₂−y₁ = x−0.5x².

V = ∫₀² 2π (3 − x) (x − 0.5x²) dx

V = ∫₀² 2π (3x − 1.5x² − x² + 0.5x³) dx

V = ∫₀² 2π (3x − 2.5x² + 0.5x³) dx

V = π ∫₀² (6x − 5x² + x³) dx

The second option is arguably easier to evaluate, but either one will get you the same answer (V = 8π/3).

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Lunna [17]

Answer:

1.  You cannot divide by zero.

2. The inequality is equivalent to -20.2 > 0, which is false.

5 0
3 years ago
Please help me with this. an explanation would be great as well
Ira Lisetskai [31]

Answer:

A

Step-by-step explanation:

test the constant 1/2 + (1/2)*(-3) = -2/2 = -1

6 0
3 years ago
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Give the function y=-5x+12, what is the output (y) when the input, x=-2?
artcher [175]

Answer:

y=22

Step-by-step explanation:

y=-5(-2)+12

y=10+12

y=22

6 0
3 years ago
How do you tell if a line is parallel perpendicular or neither?
cluponka [151]
You can tell that a line is parallel if the lines can go on forever without crossing or intersecting. A line is perpendicular when the lines intersect at a right angle. If the slopes are either equal or negative reciprocals, they cannot be parallel or perpendicular.

Hope this helps you!
3 0
3 years ago
The average price (in dollars) to rent a studio in a certain city can be approximated by the equation p - 24.1t+575 where t is
krok68 [10]

Answer:

<u>69 years</u> after 2005 which is in <u>2074</u> year the average price of a studio in this city will reach $1081.10.

Step-by-step explanation:

Given:

Average Price to rent a studio is denoted by 'p'.

It is been approximated by equation:

p-24.1t+575

where t ⇒ number of years since 2005

Now given:

Average Price to rent a studio p = $1081.10

Substituting the value of p in above equation we get;

1081.10-24.1t+575

Solving the new equation to find the value of t we get;

1656.1=24.1t

Dividing both side by 24.1 we get;

\frac{1651.1}{24.1} =\frac{24.1t}{24.1} \\\\t= 68.71 \approx 69 years

Hence<u> 69 years</u> after 2005 which is in <u>2074</u> year the average price of a studio in this city will reach $1081.10.

<u />

7 0
2 years ago
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