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ERJAEE, G. (20)

Lecture 9

Department

MathematicsCourse Code

MATH 2BProfessor

ERJAEE, G.Lecture

9This

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MATH 2B - Lecture 9 - Volume of Solids

Consider a general cylinder:

● To calculate the volume of this general solid, we cut it into many slices vertically.

● In general, for the ith slice, we use a short cylinder to approximate its volume.

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● Again, we assume each slice has the same height We use the area at the

sample point as the base.

base height

● Then we add them together to obtain the

approximation for the whole solid.

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● Then we take the limit of the above Riemann Sum:

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○ A(x): the area of intersecting face in term of x

Example:

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Find the volume obtained by rotating the region bounded by y=1, on [1,3] around

x-axis.

Solution:

● By rotating the rectangular region, we obtain a cylinder.

● Now we try to write down the definite integral for its volume:

1. Treat x as variable (since it’s rotating around the x-axis)

2. Cut the solid vertically, and find the area of intersecting plane A(x) as a

function of x.

● Cutting area is a circle with a constant radius r=1 → A(x) = πr² = π

3. Set up the integral

Example:

Compute the volume obtained by rotating the region s with x-axis which is

bounded by f(x)=x² on [0,1].

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