If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

### Course: AP®︎/College Calculus AB>Unit 8

Lesson 7: Volumes with cross sections: squares and rectangles

# Volume with cross sections: intro

Using definite integration to find volume of a solid whose base is given as a region between function and whose cross sections are squares.

## Want to join the conversation?

• Your sketches are really amazing!
• So true, noticed it ages ago, but never really considered that Sal may be a professional artist 😂😂 (just kidding)
• I am not quite sure I understand how the length of the base of the square is 6-4ln(x-3). What is the reasoning behind this?
• The length is the area under the curve between the line y=6 and and the function y=4ln(x-3). To get this, you have to subtract the area under the curve of 4ln(x-3) from the area under the curve of y=6. Hope that helps!
• Why do we need to integrate. Shouldn't we just be using summation without first finding the antiderivative?
• Hi Avi!

Integration is infinite summation. Do you remember learning derivatives using the limit definition, and then how much easier things got once you learned all the rules for differentiation? That's why we integrate... once you make the connection that infinite summation to find the net change in area is just the use of the antiderivative, it doesn't make sense to do those infinite limits (and plus, depending on the function, they can get pretty nasty).

I hope this helps a bit
:)
• I am a little confused by the area of the flat region, how can we be sure that it has the same length of 4 sides
• Since the question asks for the cross-sections to be squares, by definition all 4 side lengths of the flat region would be equal.

So, square the side length to find the area of this flat region.
• In the previous skill, what does it mean by "Calculator-active"?
Is it talking about a calculator that can take the anti-derivative of functions? Or what type of calculator is it talking about?
• I think "calculator-active" simply means the solution involves the use of a calculator.
• Isn't volume = double integration?
• You can use double integration to do volume, but this method by cross sections is much simpler to use. Double integration isn't covered until multivariable calculus.
• How can you assume they're square nut not rectangle?
• Sal states at around that any cross section will be a square, so it's given to us in the problem.
• Why is the value 6-4ln(x-3) squared?
• i am confused on whether we are suppose to square (F(X)-G(X)) in the video you square them but in the tests the answers are not squared for these
• In the video we are told that each cross section (parallel to the 𝑦-axis) of the 3-dimensional object is a square.

The side length of the cross section located 𝑥 units to the right of the 𝑦-axis is
𝑓(𝑥) − 𝑔(𝑥).

Thereby the area of this cross section is (𝑓(𝑥) − 𝑔(𝑥))².

In the practice problems the cross sections likely have other shapes and you'll have to define the area differently.