Where Mindful Learning Begins
Where Mindful Learning Begins
Free, step-by-step NCERT Solutions for both parts of this short but high-value chapter — the single exercise, 8.1, and the Miscellaneous Exercise. You'll use the definite integral from Chapter 7 to actually measure something: the area trapped between a curve and the x-axis, the area of a circle or an ellipse worked out from scratch rather than quoted from geometry, and the area under a curve that dips below the axis and needs its pieces added up carefully. Every solution here is worked the way CBSE awards marks, with the key formulas and the mistakes that cost students marks every year laid out right on this page.
This Class 12 Maths NCERT Solutions Chapter 8 hub covers Application of Integrals — the chapter that finally puts the definite integral from Chapter 7 to visible, geometric use. Elementary geometry gives you formulas for the area of a triangle, a rectangle, a circle, but it has nothing to say about the area trapped between an arbitrary curve and the x-axis. This chapter closes that gap: it slices the region into a huge number of thin strips, adds up their areas as a limit, and shows that the sum is exactly a definite integral, \int_a^b y\,dx.
The chapter is deliberately compact — there is only one numbered exercise, Exercise 8.1, which applies the idea directly to circles and ellipses in their standard form, followed by a Miscellaneous Exercise that mixes in straight lines, polynomial curves, and trigonometric curves like sin x and cos x, where the curve crosses the axis more than once. The one recurring subtlety worth mastering early is that a definite integral can come out negative when the curve dips below the x-axis, while area itself never can — so the whole chapter really comes down to slicing correctly and handling that sign carefully.
Slicing a region into thin vertical strips (∫ydx) or horizontal strips (∫xdy) turns area into a definite integral. §8.2.
When the curve dips below the x-axis, the integral goes negative — take the absolute value, and add up each piece separately. §8.2 Remark.
Using symmetry and the standard √(a²−x²) integral to derive πa² and πab from scratch. Exercise 8.1.
Splitting a region at the points where the curve crosses the axis, for lines, polynomials, and curves like sin x and cos x. Miscellaneous Exercise.
Everything you need before you start solving. This is a summary for quick recall — the Formula Cards below has the full printable version for all of Calculus.
The area bounded by the curve, the x-axis, and the ordinates x = a and x = b (b > a).
Useful whenever the curve is more naturally described as x in terms of y.
Take the absolute value of any portion below the x-axis before adding it to a portion above.
Use symmetry about both axes — find the area in one quadrant and multiply by 4.
Same symmetry trick as the circle, scaled by b/a — check which axis a and b belong to first.
Nearly every area calculation in this chapter reduces to evaluating this one standard result at its limits.
Drawn from where students actually lose marks across both parts of this chapter.
Area under simple curves, applied to the standard ellipse and circle · 4 questions
Solve Exercise 8.1 →Area under lines, polynomial curves, and trigonometric curves like sin x, including regions that cross the x-axis · 5 questions
Solve Miscellaneous →Every formula for Calculus — integrals, application of integrals — in one printable PDF.
Get Formula Cards →Quick answers about Chapter 8, Application of Integrals.
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