# Calculating the volume of the solid in this graph

• jisbon
In summary: Each cross section is perpendicular to the x-axis and has a square shape, with the sides of the square running across the base of the solid. The volume of the solid is then the sum of all these cross sections along the x-axis.
jisbon
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

jisbon said:
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.

Last edited:
Mark44 said:
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:

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

jisbon said:
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.
jisbon said:
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.

Here's a picture showing a few 3D cross sections, if that helps.

Mark44

## 1. How do you calculate the volume of a solid in a graph?

The volume of a solid in a graph can be calculated by finding the area under the curve of the graph and multiplying it by the length of the solid. This can be done using integral calculus or by using the trapezoidal rule for numerical integration.

## 2. What units are typically used to measure volume in a graph?

The units used to measure volume in a graph will depend on the units used for the length and height of the solid. For example, if the length is measured in meters and the height is measured in meters, then the volume will be in cubic meters. It is important to ensure that all units are consistent when calculating volume.

## 3. Can the volume of a solid in a graph be negative?

No, the volume of a solid in a graph cannot be negative. Volume is a measure of space and cannot be negative. If the graph appears to show a negative volume, it may be due to a miscalculation or error in the graph itself.

## 4. What if the graph is not a perfect shape, can the volume still be calculated?

Yes, the volume of a solid in a graph can still be calculated even if the graph is not a perfect shape. This can be done by dividing the solid into smaller, more manageable shapes and calculating the volume of each shape. Then, the volumes can be added together to get the total volume of the solid.

## 5. What are some real-life applications of calculating the volume of a solid in a graph?

Calculating the volume of a solid in a graph has many real-life applications, such as in engineering, architecture, and physics. It can be used to determine the volume of a container, the amount of liquid or gas that can be held in a tank, or the displacement of an object. It is also important in fields such as fluid mechanics and thermodynamics when studying the behavior of fluids in different volumes.

• Calculus and Beyond Homework Help
Replies
2
Views
1K
• Calculus and Beyond Homework Help
Replies
34
Views
2K
• Calculus and Beyond Homework Help
Replies
3
Views
1K
• Calculus and Beyond Homework Help
Replies
14
Views
930
• Calculus and Beyond Homework Help
Replies
2
Views
332
• Calculus and Beyond Homework Help
Replies
2
Views
654
• Calculus and Beyond Homework Help
Replies
10
Views
756
• Calculus and Beyond Homework Help
Replies
8
Views
3K
• Calculus and Beyond Homework Help
Replies
5
Views
798
• Calculus and Beyond Homework Help
Replies
27
Views
2K