Area Between Curves

Name: Area Between Curves
Author: StemPy Lab

A road runs between a hillside and a stream, and a surveyor wants the area of the land between them. Find where two curves meet, decide which one lies above, and integrate the difference. Then split a region where the upper curve changes, and choose vertical or horizontal strips to keep the working short. Parabolas, circles, ellipses and lines are the shapes you meet. It covers the second part of the Indian syllabus unit (NCERT, JEE Main) and the matching topic of A-Level, IB and AP.

3 lessons, about 1 h 20 min3 simulations10 problems3 practice questionsPython in every lesson

Created by StemPy Lab

Course outline (3 lessons)

Lesson 1about 25 min
Area between a curve and a line
The area between a curve and a line is the integral of the upper curve minus the lower curve, taken between the two points where they meet. Solve the two equations together to find those points. A parabola cut by a line has a shortcut: the area is $\dfrac{|a|\,d^3}{6}$ , where $a$ is the coefficient of $x^2$ and $d$ is the distance between the meeting points.
1 simulation1 practice question1 Python program3 problems
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Lesson 2about 25 min
Area between two curves
For any two curves that meet at $x = a$ and $x = b$ , with $f$ above $g$ in between, the area between them is $\displaystyle\int_a^b (f - g)\,dx$ . The formula does not care whether the curves are above or below the $x$ -axis. Two parabolas, or a parabola and a circle, are solved the same way: find the meeting points, find the upper curve, subtract.
1 simulation1 practice question1 Python program4 problems1 Python lab
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Lesson 3about 30 min
Splitting a region and choosing the strips
When two curves cross inside the range, the upper curve changes. Find every crossing and add the areas of the pieces, or integrate $|f - g|$ . Symmetry can halve the work. Sometimes horizontal strips give one integral where vertical strips give two, so choose the direction that keeps the top and the bottom of the strip the same curve.
1 simulation1 practice question1 Python program3 problems1 Python lab1 question set1 note
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Area Between Curves: Timed Test

10 questions · 25 min

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Area Between Curves

Course outline (3 lessons)

Area between a curve and a line

Area between two curves

Splitting a region and choosing the strips

Test Your Learning

Your teacher

Discussion (0)