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Calculus

Volume of Revolution

Concept Quizzes

Volume of Revolution - Disc Method
Volume of Revolution - Shell Method
Volume of Revolution - Problem Solving

Challenge Quizzes

Volume of Revolution: Level 4 Challenges

Volume of Revolution - Problem Solving

If the surface area of the hypersphere (sphere in 4 dimensions) is $2 \pi ^2 R^3$ , what would be the volume of the hypersphere?

\frac{ \pi^2 R^4}{2}

2 \pi^2 R^4

\frac{ 2 \pi^2 R^4}{3}

\frac{ 4 \pi R^4}{3 }

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by Brilliant Staff

Consider a three dimensional object whose base is a semicircle with diameter $AB = 28$ . If each cross section of the three dimensional object cut by a plane perpendicular to $\overline{AB}$ is a semicircle, as shown in the diagram above, what is the volume of the three dimensional object?

\displaystyle{{548}\pi }

\displaystyle{\frac{2744}{11}\pi}

\displaystyle{{411}\pi}

\displaystyle{\frac{1372}{3} \pi}

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by Brilliant Staff

Consider a three-dimensional shape standing on the floor with height $\frac{\pi}{2}\text{ cm}$ such that the area of a cross section parallel to the floor is $14\sin x\cdot \cos x \text{ cm}^2$ for any height $0 \le x \le \frac{\pi}{2}\text{ cm}.$ What is the volume of the three-dimensional shape in $\text{cm}^3$ ?

8 10 7 9

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by Brilliant Staff

Consider a three-dimensional shape standing on the floor with height $17 \text{ cm}$ such that the area of a cross section parallel to the floor is $\frac{3}{2} \sqrt{17-x} \text{ cm}^2$ for any height $0 \le x \le 17 .$ What is the volume of the three-dimensional shape in $\text{cm}^3$ ?

\displaystyle{{16}\sqrt{17}}

\displaystyle{{17}\sqrt{17}}

\displaystyle{{19}\sqrt{17}}

\displaystyle{{18}\sqrt{17}}

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by Brilliant Staff

In the above diagram, the equation of the curve is $y=\sqrt{ax}=\sqrt{2x}$ and the equation of the line is $y=mx.$ Let $V_x$ and $V_y$ be the volumes of the solids obtained by revolving the shaded region bounded by the curve and line about the $x$ -axis and $y$ -axis, respectively. If $V_x=V_y,$ what is the value of $m?$

\displaystyle{ \frac{4}{5}}

\displaystyle{ \frac{3}{5}}

\displaystyle{ \frac{6}{5}}

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by Brilliant Staff