Volumes of Solids 1
Try VividMath Premium to unlock full access
Time limit: 0
Quiz summary
0 of 4 questions completed
Questions:
- 1
- 2
- 3
- 4
Information
–
You have already completed the quiz before. Hence you can not start it again.
Quiz is loading...
You must sign in or sign up to start the quiz.
You have to finish following quiz, to start this quiz:
Loading...
- 1
- 2
- 3
- 4
- Answered
- Review
-
Question 1 of 4
1. Question
Find the volume generated when `y = x^3` is rotated about the `x` – axis, between `x = 0` & `x = 2`
Hint
Help VideoCorrect
Well Done!
Incorrect
Volumes-Disc Method
$$V= \pi \int_{\color{#00880A}{a}}^{\color{#9a00c7}{b}} y^2 dx$$First, make `y^2` the subject of the given equation`y` `=` `x^3` `y^2` `=` `x^6` Substitute `y^2` into the given formula and substitute the limits `x=0` and `x=2``V` `=` $$\pi \int_{\color{#00880A}{0}}^{\color{#9a00c7}{2}}$$$$y^2$$$$\:dx$$ Limits are `x=0` and `x=2` `=` $$\pi \int_{\color{#00880A}{0}}^{\color{#9a00c7}{2}}$$$$x^6$$$$\:dx$$ `y^2=x^6` Apply Power Rule`V` `=` $$\pi \int_{\color{#00880A}{0}}^{\color{#9a00c7}{2}}$$$$x^6$$$$\:dx$$ `=` $$\pi \left(\frac{x^{6+1}}{6+1} \right)$$ Apply Power Rule `=` `pi (x^7/7)` Simplify Find the Definite Integral`V` `=` $$\int_{\color{#00880A}{0}}^{\color{#9a00c7}{2}} x^6 dx$$ `=` $$\pi \left[\frac{x^7}{7}\right]_{\color{#00880A}{0}}^{\color{#9a00c7}{2}}$$ `=` $$\pi \left[\frac{\color{#9a00c7}{2}^7}{7}- \frac{\color{#00880A}{0}^7}{7}\right]$$ Substitute the upper `(2)` and lower limits `(0)` `=` `pi (128/7 – 0)` Simplify `=` `(128pi)/7` `(128pi)/7` cubic units -
Question 2 of 4
2. Question
Find the volume generated when `y = 1/2x` is rotated about the `x` – axis, between `x = 0` & `x = 4`
Hint
Help VideoCorrect
Nice Job!
Incorrect
Volumes-Disc Method
$$V= \pi \int_{\color{#00880A}{a}}^{\color{#9a00c7}{b}} y^2 dx$$First, make `y^2` the subject of the given equation`y` `=` `1/2x` `y^2` `=` `1/4x^2` Substitute `y^2` into the given formula and substitute the limits `x=0` and `x=4``V` `=` $$\pi \int_{\color{#00880A}{0}}^{\color{#9a00c7}{4}}$$$$y^2$$$$\:dx$$ Limits are `x=0` and `x=4` `=` $$\pi \int_{\color{#00880A}{0}}^{\color{#9a00c7}{4}}$$$$\frac{1}{4}x^2$$$$\:dx$$ `y^2=1/4x^2` Apply Power Rule`V` `=` $$\pi \int_{\color{#00880A}{0}}^{\color{#9a00c7}{4}}$$$$\frac{1}{4}x^2$$$$\:dx$$ `=` $$\pi \left(\frac {1}{4}\cdot\frac{x^{2+1}}{2+1} \right)$$ Apply Power Rule `=` `1/4*(pi x^3)/3` Simplify `=` `pi (x^3/12)` Find the Definite Integral`V` `=` $$\int_{\color{#00880A}{0}}^{\color{#9a00c7}{4}} \frac {x^2}{4} dx$$ `=` $$\pi \left[\frac{x^3}{12}\right]_{\color{#00880A}{0}}^{\color{#9a00c7}{4}}$$ `=` $$\pi \left[\frac{\color{#9a00c7}{4}^3}{12}- \frac{\color{#00880A}{0}^3}{12}\right]$$ Substitute the upper `(4)` and lower limits `(0)` `=` `pi (64/12 – 0)` Simplify `=` `(16pi)/3` `(16pi)/3 \text (cubic units)` -
Question 3 of 4
3. Question
Find the volume generated when `y = 2` is rotated about the `x` – axis, between `x = -3` & `x = 3`
Hint
Help VideoCorrect
Excellent!
Incorrect
Volumes-Disc Method
$$V= \pi \int_{\color{#00880A}{a}}^{\color{#9a00c7}{b}} y^2 dx$$First, make `y^2` the subject of the given equation`y` `=` `2` `y^2` `=` `4` Substitute `y^2` into the given formula and substitute the limits `x=-3` and `x=3``V` `=` $$\pi \int_{\color{#00880A}{-3}}^{\color{#9a00c7}{3}}$$$$y^2$$$$\:dx$$ Limits are `x=-3` and `x=3` `=` $$\pi \int_{\color{#00880A}{-3}}^{\color{#9a00c7}{3}}$$$$4$$$$\:dx$$ `y^2=4` Apply Power Rule`V` `=` $$\pi \int_{\color{#00880A}{-3}}^{\color{#9a00c7}{3}}$$$$4$$$$\:dx$$ `=` $$4\pi \left(\frac{x^{0+1}}{0+1} \right)$$ Apply Power Rule `=` `4pi x` Simplify Find the Definite Integral`V` `=` $$\int_{\color{#00880A}{-3}}^{\color{#9a00c7}{3}} 4 dx$$ `=` $$4\pi \left[x\right]_{\color{#00880A}{-3}}^{\color{#9a00c7}{3}}$$ `=` `4pi [\color{#9a00c7}{3} – (\color{#00880A}{-3})]` Substitute the upper `(3)` and lower limits `(-3)` `=` `4pi (6)` Simplify `=` `24pi` `24pi` cubic units -
Question 4 of 4
4. Question
Find the volume generated when `y = x^2` is rotated about the `x` – axis, between `x = 1` & `x = 3`
Hint
Help VideoCorrect
Fantastic!
Incorrect
Volumes-Disc Method
$$V= \pi \int_{\color{#00880A}{a}}^{\color{#9a00c7}{b}} y^2 dx$$First, make `y^2` the subject of the given equation`y` `=` `x^2` `y^2` `=` `x^4` Substitute `y^2` into the given formula and substitute the limits `x=1` and `x=3``V` `=` $$\pi \int_{\color{#00880A}{1}}^{\color{#9a00c7}{3}}$$$$y^2$$$$\:dx$$ Limits are `x=1` and `x=3` `=` $$\pi \int_{\color{#00880A}{1}}^{\color{#9a00c7}{3}}$$$$x^4$$$$\:dx$$ `y^2=x^4` Apply Power Rule`V` `=` $$\pi \int_{\color{#00880A}{1}}^{\color{#9a00c7}{3}}$$$$x^4$$$$\:dx$$ `=` $$\pi \left(\frac{x^{4+1}}{4+1} \right)$$ Apply Power Rule `=` `pi (x^5/5)` Simplify Find the Definite Integral`V` `=` $$\pi \int_{\color{#00880A}{1}}^{\color{#9a00c7}{3}}$$$$x^4$$$$\:dx$$ `=` $$\pi \left[\frac{x^5}{5} \right]_{\color{#00880A}{1}}^{\color{#9a00c7}{3}}$$ `=` $$\pi \left[\frac{\color{#9a00c7}{3}^5}{5} – \frac{\color{#00880A}{1}^5}{5}\right]$$ Substitute the upper `(3)` and lower limits `(1)` `=` `pi (243/5-1/5)` Simplify `=` `(242/5)pi` `=` `(242pi)/5` `(242pi)/5` cubic units