Area Under a Curve

Name: Area Under a Curve
Author: StemPy Lab

How much paint covers a wall with a curved top? Cut the region into thin strips, add the areas of the strips, and let the strips become thinner and thinner. The sum becomes a definite integral. You find the area under a curve, the area measured along the y-axis and the area of a region where the curve crosses the axis. The line, the parabola, the circle and the ellipse are the standard cases. It covers the first 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 30 min
Area as the limit of strips
To find the area of a region with a curved edge, cut it into thin strips and treat each strip as a rectangle. Add the areas of the strips. As the strips get thinner, the sum settles on the exact area. That limit is the definite integral, and an antiderivative gives it without any limit.
1 simulation1 practice question1 Python program3 problems1 Python lab
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Lesson 2about 25 min
Area under a curve and along the y-axis
The area bounded by $y = f(x)$ , the $x$ -axis and two vertical lines is $\int_a^b y\,dx$ . When the curve is given as $x = g(y)$ , cut the region into horizontal strips instead, and the area bounded by the curve, the $y$ -axis and two horizontal lines is $\int_c^d x\,dy$ . The same rules give the areas of a line, a parabola, a circle and an ellipse.
1 simulation1 practice question1 Python program3 problems
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Lesson 3about 25 min
Curves that cross the axis
The integral of a function counts a part below the $x$ -axis as negative, so it gives the signed area. The total area counts every part as positive. To find it, split the interval at the points where the curve meets the axis, integrate each piece, and add the absolute values.
1 simulation1 practice question1 Python program4 problems1 Python lab1 question set1 note
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Test Your Learning

Area Under a Curve: Practice Paper

15 questions · 20 min

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StemPy Lab

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Area Under a Curve

Course outline (3 lessons)

Area as the limit of strips