Integral Calculus Questions and Answers

71.

Put these steps of finding a volume of revolution in order. 1. Square the radius and multiply by \( \pi \) 2. Integrate between the limits 3. Identify the axis and the radius of a typical slice 4. Sketch the region and fix the limits of integration

A.3, 4, 1, 2
B.4, 1, 3, 2
C.4, 3, 2, 1
D.4, 3, 1, 2

Answer: D

The sketch settles which variable runs and between what values, so it comes first. The axis then determines what counts as the radius of a slice, which is squared and scaled to give the slice volume. Integration is the last step, and attempting it before the radius is identified is what produces the wrong power of \( y \)

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72.

Three of these four hold for the definite integral. The exception is

A.it is a number rather than a function
B.it changes sign when the limits are exchanged
C.it depends on the constant of integration chosen
D.it is additive over adjoining intervals

Answer: C

The constant cancels in \( F(b) - F(a) \), so it cannot affect the value: this is precisely why a definite integral is well defined although the antiderivative is not unique. Additivity, being a number, and the sign change on exchanging limits are all genuine properties

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73.

None of these four is a valid substitution for \( \int \sqrt{a^{2} - x^{2}}\,dx \) except

A.\( x = a\cosh u \)
B.\( x = a\sin\theta \)
C.\( x = a\tan\theta \)
D.\( x = a\sec\theta \)

Answer: B

With \( x = a\sin\theta \) the radical becomes \( a\cos\theta \), removing the root cleanly. The tangent substitution suits \( a^{2} + x^{2} \), and the secant and hyperbolic cosine suit \( x^{2} - a^{2} \); each of those three leaves the wrong sign under the root here and produces an imaginary result

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Integral Calculus Questions and Answers

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Integral Calculus Questions and Answers

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