Volumes of reolution

Draw

Draw a right-angled triangle against the x-axis. On top of what you have drawn now draw the 3D shape it would form when rotated around the x-axis.

Open Ended

What is the name of the 3D shape that was created in the previous question?

Multiple Choice

What is formed when the area between the function and the x-axis is rotated through 2π about the x-axis?

• A solid of revolution

• A flat shape

• A 2D figure

• A line

Multiple Choice

• $$\int_1^a\pi\left(\frac{1}{\sqrt[]{x}}\right)^{ }dx$$  

• $$\int_1^a\pi\left(\frac{1}{\sqrt[]{x}}\right)^2$$  

• $$\int_1^a\pi\left(\frac{1}{\sqrt[]{x}}\right)^2dx$$  

• $$\int_1^a\left(\frac{1}{\sqrt[]{x}}\right)^2dx$$  

Multiple Choice

Use integration to find the volume of the solid generated when the line y = x for 1 ≤ x ≤ 4 is revolved through 2π around the x-axis. What is the volume of the solid?

• 21π units³

• 15π units³

• 7.5π units³

• 30π units³

Draw

Draw a sketch of the 3D shape formed in the previous question

Open Ended

Find the volume of the solid formed when the graph of the function y = x² for 0 ≤ x ≤ 5 is revolved through 2π about the x-axis.

Draw

Draw a sketch of the 3D shape formed in the previous question

Multiple Choice

What is the volume of revolution when the graph of y = ln(x) for x ∈ [1, e] is revolved about the y-axis?

• π(e^2 - 1)/2 units³

• π(e - 1)/2 units³

• π(e^2 + 1)/2 units³

• π(e^2 - 2)/2 units³

Draw

Draw a sketch of the 3D shape formed in the previous question

Draw

Starting with the double angle identity cos(2x)≡cos²x−sin²x, and by using the Pythagorean identity sin²x+cosx≡1, prove that:

sinx=1∕2​(1−cos(2x))

Math Response

One arch of y=sinx is revolved through 2π radian about the x-axis. Find the volume of revolution.

Type answer here

Correct Answer: \pi^2\div2

Deg^°

Rad

Multiple Choice

What do you think the correct formula should be for the volume of the solid of revolution created by revolving the area between two functions?

• V = π∫[f(x)]² dx

• V = π∫[g(x)]² dx

• V = π∫[f(x) - g(x)]² dx

• V = π∫[f(x)² - g(x)²] dx

Open Ended

Find the volume of revolution generated by revolving the region between y = x² and y = √x about the x-axis.

Poll

How confident do you feel about this topic now?

• Very confident
• Somewhat confident
• Not confident