Calculating the volume of the solid in this graph

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Homework Statement:
Base of solid is the region bounded by graphs ##y= \sqrt x## and ##y=x/2##. The cross sections perpendicular to the x axis are squares whose sides run across the base of the solid. Find volume of solid.
Relevant Equations:
-
Homework Statement: Base of solid is the region bounded by graphs ##y= \sqrt x## and ##y=x/2##. The cross sections perpendicular to the x axis are squares whose sides run across the base of the solid. Find volume of solid.
Homework Equations: -

As stated above, I will want to calculate the coordinates of the two graphs intersecting, and found them to be x=0 and x=4. However, the equation ## \pi \int_{0}^{4} (\sqrt x - x/2) ^2##seems to be wrong. Referring to my answer key, the answer was simply## \int_{0}^{4} (\sqrt x - x/2)^2## without the pi. Why is this so? Thanks
 

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  • #2
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Homework Statement: Base of solid is the region bounded by graphs ##y= \sqrt x## and ##y=x/2##. The cross sections perpendicular to the x axis are squares whose sides run across the base of the solid. Find volume of solid.
Homework Equations: -

As stated above, I will want to calculate the coordinates of the two graphs intersecting, and found them to be x=0 and x=4. However, the equation ## \pi \int_{0}^{4} (\sqrt x - x/2) ^2##seems to be wrong. Referring to my answer key, the answer was simply## \int_{0}^{4} (\sqrt x - x/2)^2## without the pi. Why is this so? Thanks
In the future, please post problems that involve derivatives, integrals, and other calculus topics in the Calculus & Beyond section, not in the Precalc section. I will move this thread.

Regarding the problem, why do you have ##\pi## in your integrals? Nothing is being revolved around any axis -- the problem is just a geometric object whose base is a sort of crescent shape.

Draw a sketch of the region, and another sketch of the solid with a few of the vertical slices. Your textbook might even have a drawing of the object, or at least show one or two of the vertical cross sections.
 
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  • #3
467
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In the future, please post problems that involve derivatives, integrals, and other calculus topics in the Calculus & Beyond section, not in the Precalc section. I will move this thread.

Regarding the problem, why do you have ##\pi## in your integrals? Nothing is being revolved around any axis -- the problem is just a geometric object whose base is a sort of crescent shape.

Draw a sketch of the region, and another sketch of the solid with a few of the vertical slices. Your textbook might even have a drawing of the object, or at least show one or two of the vertical cross sections.
Sorry. I will take note of this in the future.

As for the question, I don't really understand what the question means by:" The cross-sections perpendicular to the x axis are squares whose sides run across the base of the solid. Find the volume of solid. "

Sketching the graphs as shown:

1571052679586.png


Am I just supposed to find the following shaded area in green? What about the second sentence?

1571052729952.png
 
  • #4
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Am I just supposed to find the following shaded area in green?
No, you're supposed to find the volume of the solid whose base is the green region.
What about the second sentence?
The region in green is the base of the solid, as viewed from above. Above the green region are squares whose width equals the differences between the curve and the line. At both the left end and right end, where the curve and line meet, the distance between the curve and line is zero, so the squares would really be just a single point. When x = 1, the vertical distance between the curve and the line is 1/2, so at that point the height of the square would also be 1/2. At x = 2, the distance between the curves is about .4, so the height of the square there would be about .4. To find the volume of the solid, think about making a bunch of slices along vertical lines, with the slices being ##\Delta x## in width. A typical volume element would be ##A(x) \cdot \Delta x##, with A(x) being the area of a particular slice.
 
  • #5
LCKurtz
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Here's a picture showing a few 3D cross sections, if that helps.
sections.jpg
 

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