Region R is enclosed by the curves and

MATHEMATICS · Single correct · CBSE · Mathematics · Applications of Integrals · Applications of Integrals

Problem

Region R is enclosed by the curves

x=y^{2}

and

x=6-y^{2}

. To compute the area of R, a student may integrate either with respect to

x

or with respect to

y

. Which of the following best explains why integrating with respect to

y

is more efficient? (a) Integrating with respect to

y

automatically accounts for the symmetry of the region, so only one side needs to be considered, whereas integrating with respect to

x

would require doubling the result. (b) Horizontal slices (integrating with respect to

y

) give a single integral with a simple polynomial integrand, whereas vertical slices (integrating with respect to

x

) would require splitting the region into two integrals that involve square–root expressions. (c) The limits of integration are easier to identify in terms of

y

because they are simple integers, whereas the corresponding

x

–limits are irrational numbers. (d) The antiderivative of the integrand expressed in

x

is not obtainable using elementary functions, while the integrand expressed in

y

is a simple polynomial.

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