Lesson Finding the volume of a solid body mentally

This Lesson (Finding the volume of a solid body mentally) was created by by ikleyn(52942)

: View Source, Show
About ikleyn:

Finding the volume of a solid body mentally

Problem 1

What is the volume of the solid in xyz-space bounded by surfaces y = x^2, y = 2 - x^2, z = 0 and z = y + 3?

Solution

I will show how to solve this problem mentally.

In (x,y)-plane, the area concluded between y = x^2, x-axis, and 0 <= x <= 1 is 1/3.

It is elementary calculation from Calculus.


From it, we deduce, that the area concluded between y = x^2, y = 1  and 0 <= x <= 1 is 2/3.


Hence, the area concluded between y = x^2 and y = 2-x^2 is 4 times 2/3, or 8/3 square units.


Now, our 3D solid consists of two parts.


One part is a right cylinder 0 <= z <= 3 over its base, which the area concluded 
between y = x^2 and y = 2-x^2.


The volume of this cylinder is   = 8 cubic units.



The other part is half of the cylinder  3 <= z <= 5  with the same base.

The height of this imaginary cylinder is 2 units (z from 3 to 5), so, its volume is  = 16/3.


The plane z = z + 3 cuts this cylinder in two parts of equal volumes - it is clear from the symmetry.


So, the whole volume of the 3D body under the interest is   =  =  = 32/3 cubic units.

My other lessons on Calculus word problems at this site are
    - A ladder foot slides on the ground
    - Finding rate of change of some processes
    - Find the derivative of a function defined by complicated expression
    - Taking derivative of a function, which is defined implicitly
    - Find the derivative for a function satisfying given functional equation
    - Find the range of f(x) = (5*cos(x))/(x + 1)), x >=0
    - A tricky Calculus problem on derivative and anti-derivative
    - Couple of non-standard Calculus problems
    - Finding the minimum of a function defined on a curve in a coordinate plane
    - Tricky solution to find the maximum of a function defined by a complicated expression
    - Maximize the area of a trapezoid
    - Maximize the volume of an open box
    - Minimize surface area of a rectangular box with the given volume
    - Minimize the cost of an aquarium with the given volume
    - Minimize surface area of a conical paper cup with the given volume
    - Find the volume of a solid obtained by rotation of some plane shape about an axis
    - OVERVIEW of my lessons on Calculus word problems

Use this file/link ALGEBRA-II - YOUR ONLINE TEXTBOOK to navigate over all topics and lessons of the online textbook ALGEBRA-II.

This lesson has been accessed 306 times.