Applications of Integrals
A farmer wants to know how much land lies between a stream and a fence. A builder wants the volume of water in a curved tank. Both questions ask for a total made of many thin pieces, and the integral adds those pieces. This chapter uses the integral to find areas under and between curves, the volume of a solid made by turning a curve, the average value of a function and the distance from a velocity graph. It covers the unit of the Indian syllabus (NCERT, JEE Main) and the matching topics of A-Level, IB and AP. Python checks every area on the computer.
Part of Mathematics · High School (11–12) (chapter 12)
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Area Under a Curve
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.
Area Between Curves
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.