Volume of a solid bound by region work shown

In summary, the problem is to find the volume of a solid bounded by y= 2*sqrt(sin(x)) and the x-axis, with x an element of [0, (pi/2)], where the cross sections perpendicular to the x-axis are squares. The approach is to use the formula A(x) = [sqrt(sin(x))]^2 for the area of each cross section, and integrate it from 0 to pi/2 to find the total volume. However, there may be an error in the approach and further assistance is needed.
  • #1
johnq2k7
64
0
The base of a solid is the region bounded by y= 2*sqrt(sin(x)) and the x-axis, with x an element of [0, (pi/2)]. Find the volume of the solid, given that the cross sections perpendicular to the x-axis are squares.

Work Shown:

cross sections are squares:

therefore A(x) is not equal to Pi*r^2 rather l^2

therefore A(x)= [sqrt(sin(x)]^2

therefore integral of A(x)dx of x-values from 0 to Pi/2 should provide the answer

I think my approach is wrong and my integral for A(x) dx is wrong as well please help!
 
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  • #2
Hi johnq2k7! :smile:

(have a pi: π :wink:)

Seems ok (except you seem to have droppoed the original 2) …

the general rule is to specify the slices you're dividing it into … in this case, slices of thickness dx and area y2. :wink:
 

FAQ: Volume of a solid bound by region work shown

What is the formula for finding the volume of a solid bound by a region?

The formula for finding the volume of a solid bound by a region is to integrate the cross-sectional area of the region with respect to the axis of rotation. This can be expressed as V = ∫A(x)dx, where A(x) is the cross-sectional area at a given point along the axis of rotation and dx represents an infinitely small element along the axis.

How do you determine the cross-sectional area of a region?

The cross-sectional area of a region can be determined by using basic geometry or by using mathematical methods such as integration. For simple shapes like rectangles, triangles, and circles, the cross-sectional area can be calculated using their respective formulas. For more complex shapes, integration is used to break down the shape into small, infinitesimal elements and summing their areas to find the total cross-sectional area.

What is the difference between a solid bound by a region and a regular solid?

A solid bound by a region is a three-dimensional object that is formed by rotating a two-dimensional region about an axis, whereas a regular solid is a three-dimensional object with a uniform shape and size. A solid bound by a region can have varying shapes and sizes depending on the region and axis of rotation, while a regular solid has a fixed shape and size.

How is the volume of a solid bound by a region related to the work done?

The volume of a solid bound by a region can be related to the work done by understanding that the volume represents the amount of space occupied by the object. When work is done on the object, such as rotating it about an axis, the object's energy and shape change, which affects its volume. Therefore, the volume calculation is necessary to determine the work done on the object.

Can the volume of a solid bound by a region be negative?

No, the volume of a solid bound by a region cannot be negative. Since volume is a measure of space, it cannot be less than zero. If the calculated volume is negative, it is an indication that a mistake was made in the calculation or the chosen region and axis of rotation do not form a solid object.

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