What traumatic event does Zitkala-Sa describe at the school?

The forcible cutting of her long hair

Why was the cutting of her hair so humiliating to Zitkala-Sa?

Among her people, shingled hair was worn by cowards and mourners

Write the area under \(y=f(x)\) from \(x=a\) to \(x=b\).

The area is \(\int_a^b |f(x)|\,dx\); the modulus ensures regions below the axis count positively.

Write the area between curves \(y=f(x)\) and \(y=g(x)\).

Over \([a,b]\) with \(f\geq g\), the area is \(\int_a^b [f(x)-g(x)]\,dx\).

How are the limits of integration determined for area between curves?

By solving \(f(x)=g(x)\) to find the \(x\)-coordinates of the points where the curves intersect.

How is area computed with respect to the \(y\)-axis?

Express \(x\) as a function of \(y\), \(x=g(y)\), and integrate \(\int_c^d |g(y)|\,dy\).

Find the area of a circle \(x^2+y^2=a^2\) by integration.

By symmetry it is \(4\int_0^a \sqrt{a^2-x^2}\,dx=\pi a^2\).

Why use the modulus in the area formula?

Because the integral of a function below the \(x\)-axis is negative, while geometric area must be positive; the modulus corrects the sign.

Set up the area of the region bounded by \(y^2=4ax\) and a line.

Find intersection points to fix limits, then integrate the difference of the right boundary and left boundary over those limits.

What is the area under \(y=\sin x\) from \(0\) to \(\pi\)?

\(\int_0^{\pi} \sin x\,dx=[-\cos x]_0^{\pi}=2\).

How do you handle a curve that crosses the \(x\)-axis within \([a,b]\)?

Split the integral at the crossing point and take the absolute value of each piece, then add.

Write the area of an ellipse \(\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1\).

By integration the area is \(\pi a b\).

Application of Integrals

Class 12 Mathematics · Mathematics Part II · Chapter 2

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CBSE Class 12 · Mathematics

Integrals: 35 marks across the four papers

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CBSE Class 12 · Mathematics

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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:

Flashcards & interactive practice