Application of Integrals
Class 12 Mathematics · Mathematics Part II · Chapter 2
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Summary
This chapter uses the definite integral to compute areas of plane regions. The area bounded by a curve \(y=f(x)\), the \(x\)-axis and the lines \(x=a\) and \(x=b\) is \(\int_a^b |f(x)|\,dx\); similarly, area with respect to the \(y\)-axis uses \(\int_c^d |g(y)|\,dy\). The area between two curves \(y=f(x)\) and \(y=g(x)\) over \([a,b]\) is \(\int_a^b |f(x)-g(x)|\,dx\). Standard applications include the area of regions bounded by circles, parabolas, ellipses and straight lines. Care is taken with the sign of the integrand — areas below the axis contribute positively in absolute value — and with finding the correct limits from the points of intersection of the curves. These techniques translate the abstract integral into concrete geometric measurement.
Area under simple curvesArea bounded by a curve and a lineArea between two curvesAreas of circles, parabolas and ellipses
Key terms
Area under a curve
\(\int_a^b |f(x)|\,dx\) for \(y=f(x)\) between \(x=a\) and \(x=b\).
Area between two curves
\(\int_a^b |f(x)-g(x)|\,dx\).
Limits of integration
The \(x\)-values where the bounding curves intersect or are specified.
Area w.r.t. y-axis
\(\int_c^d |g(y)|\,dy\) using \(x=g(y)\).
Region of integration
The bounded plane region whose area is being computed.
Signed area
The integral can be negative; geometric area uses its absolute value.
Extra questions & answers
Try one from this chapter
The area under \(y=f(x)\) between \(x=a\) and \(x=b\) is: