MATHEMATICS HIGH SCHOOL

Refer to the figure and find the volume V generated by rotating the given region about the specified line.R3 about AB
Refer to the figure and find the volume V generated - 1

Answers

Answer 1
Answer:

Answer:

Hence, volume is: (34\pi)/(45) cubic units.

Step-by-step explanation:

We will first express our our equation of the curve and the line bounded by the region in terms of the variable y.

i.e. the curve is rex=(1)/(16)y^4

and the line is given as:  x=(1)/(2)y

Since after rotating the given region R_(3) about the line AB.

we see that for the following graph

the axis is located at x=1.

and the outer radius(R) is: (1)/(16)y^4

and the inner radius(r) is:  (1)/(2)y

Now, the area of the graph= area of the disc.

Area of graph=\pi(R^2-r^2)

Now the volume is given as:

Volume=\int\limits^2_0 {Area} \, dy

On calculating we get:

Volume=(34\pi)/(45) cubic units.
Answer 2
Answer:

The volume V generated by rotating the given region about the specified line R3 about AB is \boxed{\frac{{34\pi }}{{45}}{\text{ uni}}{{\text{t}}^3}}.

Further explanation:

Given:

The coordinates of point A is \left( {1,0} \right).

The coordinates of point B is \left( {1,2} \right).

The coordinate of point C is \left( {0,2} \right).

The value of y is y = 2\sqrt[4]{x}.

Explanation:

The equation of the curve is y = 2\sqrt[4]{x}.

Solve the above equation to obtain the value of x in terms of y.

\begin{aligned}{\left( y \right)^4}&={\left( {2\sqrt[4]{x}} \right)^4} \n{y^4}&=16x\n\frac{1}{{16}}{y^4}&= x\n\end{aligned}

The equation of the line is x = (1)/(2)y.

After rotating the region {R_3} is about the line AB.

From the graph the inner radius is {{r_2} = (1)/(2)y and the outer radius is {{r_1}=\frac{1}{{16}}{y^4}.

{\text{Area of graph}}=\pi\left( {{r_1}^2 - {r_2}^2} \right)

Area = \pi\left( {{{\left({\frac{1}{{16}}{y^4}} \right)}^2} - {{\left({(1)/(2)y} \right)}^2}}\right)

The volume can be obtained as follows,

\begin{aligned}{\text{Volume}}&=\int\limits_0^2 {Area{\text{ }}dy}\n&=\int\limits_0^2{\pi \left( {{{\left({\frac{1}{{16}}{y^4}} \right)}^2} - {{\left( {(1)/(2)y} \right)}^2}} \right){\text{ }}dy}\n&= \pi \int\limits_0^2 {\left( {\frac{1}{{256}}{y^8} - (1)/(4){y^2}} \right){\text{ }}dy}\n\end{aligned}

Further solve the above equation.

\begin{aligned}{\text{Volume}}&=\pi \left[ {\int\limits_0^2 {\frac{1}{{256}}{y^8}dy - } \int\limits_0^2{(1)/(4){y^2}{\text{ }}dy} } \right]\n&= \frac{{34\pi }}{{45}}\n\end{aligned}

The volume V generated by rotating the given region about the specified line R3 about AB is \boxed{\frac{{34\pi }}{{45}}{\text{ uni}}{{\text{t}}^3}}.

Learn more:

1. Learn more about inverse of the functionbrainly.com/question/1632445.

2. Learn more about equation of circle brainly.com/question/1506955.

3. Learn more about range and domain of the function brainly.com/question/3412497

Answer details:

Grade: High School

Subject: Mathematics

Chapter: Volume of the curves

Keywords: area, volume of the region, rotating, generated, specified line, R3, AB, rotating region.

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Answers

If you would like to know how many pine and maple trees will be planted, you can calculate this using the following steps:

2 pine trees + 3 maple trees = 5 trees

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Answer:

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Step-by-step explanation:

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Step-by-step explanation:

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Answers

Answer:

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Step-by-step explanation:

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--------

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2) 47

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---------

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3) 3 8

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----------

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-----------

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 ___

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2)47

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Answers

Answer:

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Step-by-step explanation:

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