The trapezium rule gives $0.76$ for $\int_0^1x^2e^x\,dx$ , whose exact value is $e-2\approx0.718$ . Is this an over-estimation or under-estimation, and why?

AAn over-estimation, since

x^2e^x

is convex (curves upward) on

[0,1]

, so the chord segments used by the trapezium rule lie above the curve.

BAn over-estimation, but only because too few ordinates (5) were used; more ordinates would make it an under-estimation instead.

CAn under-estimation, since

x^2e^x

is concave (curves downward) on

[0,1]

, so the chord segments lie below the curve.

DNeither: the trapezium rule always gives the exact value for any smooth, continuous function.

Explanation: Since

0.76>0.71828

, the trapezium rule over-estimates the integral here, consistent with

x^2e^x

being convex on

[0,1]

, which makes each trapezium's chord lie above the actual curve.

Derived from ZIMSEC Pure Mathematics Paper 2, November 2025, Q10

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