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Course: AP®︎/College Calculus AB>Unit 8

Lesson 7: Volumes with cross sections: squares and rectangles

Volume with cross sections: squares and rectangles (no graph)

Expressing the volume of a 3-D figure as a definite integral (where the area of cross sections are functions of x). The problem doesn't come with a graph but that doesn't mean we shouldn't sketch one!

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• how is x=3 fox vertex how did we arrive at that conclusion?
• Well the vertex is (3,8). The reason we know the vertex is (3,8) is because we found the two points of intersection to be when x=1 and x=5. Since parabolas (the graphs of quadratic functions) are symmetrical, the axis of symmetry would occur when x=3, the x-value halfway between 1 and 5. When you substitute 3 into the quadratic function, you get 8 for the output. So, the vertex is (3,8).
• I evaluated this definite integral by hand and by calculator and got 32 units as the volume. Is this correct?
• Yes. Plugging the definite integral into a calculator yields 32 cubic units.
• At , why do we multiply the function by x? I know how to do everything, but that is confusing me.
• We are multiplying the base times height to find the area of the cross section. Since the base is -x^2+6x-5 and then you must multiply it by the given height which is x
• Hi, I don't know where to ask this, but how do I turn off the "subtitles/closed captions" on all videos. Do I have to do this on all videos that I am playing?
• The subtitles appear to due 'Closed Captions'(CC) being on. If you turn off CC which is at the bottom of the screen you can watch video without subtitles. The alternative would be click on the subtitle than drag so it is out of the way.
• I see how you did this using rectangle cross sections, but how do you do the square one? The area of the base would be the region squared right?
• Yes, if it's square then region squared.
• why do u multiply by x again?
• Shouldn't the height be y? I am not sure why Sal used x as the height.
• Why sir Sal only considered the upper enclosure? there are two enclosures right? The first one is in the video while the unmentioned one is the region where it is with respect to x-axis
(1 vote)
• The first slice that Khan drew @x=2 is almost like a trapezoid. He should have used the height as x=2 on both sides to make it look like a rectangle. The one he drew @x=5 is good because both the height was x.
(1 vote)
• at why is the height of the second triangle much larger than the first? I thought the height was given as x does it not have to be constant?
(1 vote)
• I think that x is the value of the graph (it is not constant). For example, at (1,4), the x value is 1 and at (5,4), the x value is 5.

I hope this helps! :D
(1 vote)