Volumes, Averages and Other Uses of the Integral

Name: Volumes, Averages and Other Uses of the Integral
Author: StemPy Lab

Turn a curve around an axis and it sweeps out a solid, like a bowl or a vase made on a lathe. Find its volume with thin discs and washers. Then find the average value of a function, the distance travelled from a velocity graph and the total of other rates, and end with the length of a curve. The Indian syllabus stops at area. These topics belong to A-Level Further Maths, IB Higher Level and AP Calculus, and they use the same idea of adding thin pieces.

3 lessons, about 1 h 35 min5 simulations10 problems5 practice questionsPython in every lesson

Created by StemPy Lab

Course outline (3 lessons)

Lesson 1about 35 min
Volume of a solid of revolution
Turn a region about an axis and it sweeps out a solid. Cut the solid into thin discs, each of volume $\pi y^2\,dx$ , and add them. This gives $V = \pi\displaystyle\int_a^b y^2\,dx$ about the $x$ -axis, and the same with $x^2\,dy$ about the $y$ -axis. A region with a hole gives washers.
2 simulations2 practice questions1 Python program4 problems1 Python lab
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Lesson 2about 35 min
Mean value and accumulation
The mean value of a function on $[a, b]$ is $\dfrac{1}{b - a}\displaystyle\int_a^b f(x)\,dx$ , the height of the rectangle with the same area. The integral of a rate over a time gives the total change, so the integral of velocity is displacement, and the integral of speed is distance.
2 simulations2 practice questions2 Python programs4 problems
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Lesson 3about 25 min
The length of a curve
A curve is nearly straight over a tiny piece. The piece has length $ds = \sqrt{dx^2 + dy^2} = \sqrt{1 + (dy/dx)^2}\,dx$ . Adding the pieces gives $L = \displaystyle\int_a^b \sqrt{1 + [f'(x)]^2}\,dx$ . This is an enrichment lesson: the Indian syllabus does not ask for it.
1 simulation1 practice question1 Python program2 problems1 Python lab1 question set1 note
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Volumes, Averages and Other Uses of the Integral: Chapter Test

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Volumes, Averages and Other Uses of the Integral

Course outline (3 lessons)

Volume of a solid of revolution

Mean value and accumulation

The length of a curve

Test Your Learning

Your teacher

Discussion (0)