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## AP®︎/College Calculus BC

### Course: AP®︎/College Calculus BC>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!

## Want to join the conversation?

• 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.
• 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.
• 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