How would I go about finding the volume of pool that is Length 10ft by Width 8ft?

firestopper53sh

firestopper53sh

Open question

2022-08-19

Double Integrals and finding the volume of a pool
I am trying to find the volume of a pool and Im a little rusty on the calculus. How would I go about finding the volume of pool that is Length 10ft by Width 8ft? The depth goes from 3 to 6 feet with a slope of 3ft/10ft.

Answer & Explanation

Andre Ferguson

Andre Ferguson

Beginner2022-08-20Added 12 answers

Step 1
The pool is a cuboid of length 10 ft breadth 8 ft and height 3 ft placed over an inverted wedge of dimensions base length 10 ft, base breadth 8 ft and height 3 ft.
Thus, volume of the pool is the volume of the cuboid plus the volume of the inverted wedge i.e,
V pool = V cuboid + V inverted wedge
( 8 × 10 × 3 ) + ( 1 2 × 8 × 10 × 3 )
= 360  ft 3
I guess if you really wanted it in terms of integration then we can derive these volumes via integration.
Step 2
The area of the cross section of the wedge is given by
0 8 3 8 y d y
And the volume will be the integration of this cross section over the base length i.e.
V inverted wedge = 0 10 ( 0 8 3 8 y d y ) d x
You can do the same for a cuboid. You should get
V cuboid = 0 10 ( 0 8 3 d y ) d x

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