Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2025 / Paper 2 / 15 / a

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- Solution a

Solution

Use the given Cartesian coordinates and integrate with respect to

y

first. Since

cos y e^{s i n y}

is the derivative of

e^{s i n y}

, the double integral becomes

I = \int_{0}^{π} cos x [e^{s i n y}]_{y = 0}^{y = x} d x = \int_{0}^{π} cos x (e^{s i n x} - 1) d x .

A direct antiderivative avoids any issue with the nonmonotonicity of

sin x

on this interval:

I = [e^{s i n x} - sin x]_{0}^{π} = 0 .

The sign changes in

cos x

make this cancellation possible even though

e^{s i n y}

is positive.

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