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Finding Volumes of Solids with Given Base Shapes and Cross-Sectional Areas, Lecture notes of Calculus

Exercises for calculating the volumes of solids with different base shapes and cross-sectional areas. The base shapes include squares, semicircles, and an equilateral triangle. The cross-sectional areas are determined by planes perpendicular to the x-axis. Each exercise includes a region defined by the graphs of two functions and instructions for finding the volume of the solid with that region as its base, given the shape of the cross-sections.

Typology: Lecture notes

2021/2022

Uploaded on 09/27/2022

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VOLUMESBYCROSSSECTIONS
Givenasolid,boundedbytwoparallelplanes
perpendicularto
x
‐axisat
x
=
a
and
x
=
b
,whereeach
cross‐sectionalareaisperpendiculartothe
x
‐axis.
CROSSSECTIONSTAKEN
PERPENDICULARTOY‐AXIS
CROSSSECTIONSTAKEN
PERPENDICULARTOX‐AXIS
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VOLUMES BY CROSS SECTIONS

Given a solid, bounded by two parallel planes

perpendicular to x ‐axis at x = a and x = b , where each

cross‐sectional area is perpendicular to the x ‐axis.

CROSS SECTIONS TAKEN

PERPENDICULAR TO Y‐AXIS

CROSS SECTIONS TAKEN

PERPENDICULAR TO X‐AXIS

Let (^) R be the region bounded by the graphs of and. Find the volume of the solid that has R as its base if every cross section by a plane perpendicular to the x ‐ axis has the given shape.

EX #1: A SQUARE

EX #3: An EQUILATERAL TRIANGLE

EX #4:

Let R be the region bounded by the graphs of

and. Find the volume of the solid that has R as its base

if every cross section by a plane perpendicular to the x ‐axis

are rectangles for which the height is four times the base

EX #6: Find the volume of the solid whose base is the region inside the circle if the cross sections taken perpendicular to the y ‐axis are squares..