Volume of Revolution

Let $V$ be the volume of the solid obtained by revolving the curve $y = x^2$ from $x = 2$ to $3$ about the $x$ -axis. What is $10 V?$

Let $V$ be the volume of the solid obtained by revolving the curve $x^2+y^2 -100 =0$ from $x = 0$ to $x=4$ about the $x$ -axis. What is $3 V?$

If $a$ is a positive number such that the volume of the solid obtained by rotating the ellipse ${x}^2 + a{y}^2 = 1$ around the $x$ -axis is $\frac{4}{57}\pi ,$ what is $a?$

The yellow-colored region in the above diagram is bounded by $\begin{array}{c}&y= 18\sin x - a \ (0 < a \ < 18), &x=0, &x=\pi, &y=0 .\end{array}$ What is the value of $a$ that minimizes the volume of the solid obtained by rotating the region around the $x$ -axis?

The volume of the solid obtained by rotating the region bounded by $y = x^2 - 2x$ and $y = x$ about the line $y = 6$ , has the form $\frac {a}{b} \pi$ , where $a$ and $b$ are positive coprime integers. What is the value of $a+b$ ?

Concept Quizzes

Challenge Quizzes

Volume of Revolution - Disc Method

Excel in math and science

Master concepts by solving fun, challenging problems.

It's hard to learn from lectures and videos

Learn more effectively through short, interactive explorations.

Used and loved by over 7 million people

Learn from a vibrant community of students and enthusiasts, including olympiad champions, researchers, and professionals.