Answer: f(c) × integral from a to b of g(x) dx for some c in [a,b].
- A f(c) × integral from a to b of g(x) dx for some c in [a,b]
- B f(a) × G(b) + f(b) × [G(b)-G(a)], an integration-by-parts-style expansion
- C The product of integral f(x) dx and integral g(x) dx taken separately
- D Cannot be simplified beyond the original product integral
Correct answer: A. f(c) × integral from a to b of g(x) dx for some c in [a,b]
Explanation: Second MVT for integrals: if f is monotone and g is integrable, then integral fg = f(c) integral g for some c. Generalization allows Bonnet form.
The exact area under the curve from a to b (shaded) is approximated by a few rectangles, the Riemann sum idea behind the definite integral.