×
Calculus

# Volume of Revolution - Disc Method

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 $\begin{array} &y= 18\sin x - a \ (0 < a \ < 18), &x=0, &x=\pi, &y=0 .\end{array}$ 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 $$a+b$$?

×