If the trapezoidal method is used to evaluate the integral ∫_0^1 x^2 dx, then…
2005
If the trapezoidal method is used to evaluate the integral ∫_0^1 x^2 dx, then the value obtained
- A.
is always > 1/3
- B.
is always < 1/3
- C.
is always = 1/3
- D.
may be greater or lesser than 1/3
Show answer & explanation
Correct answer: A
The exact value of the integral is:
∫_0^1 x^2 dx = [x^3/3]_0^1 = 1/3.
The trapezoidal method approximates the area under a curve by joining function values with straight line segments. For f(x) = x^2, the curve is convex on [0,1] because f''(x) = 2 > 0.
For a convex function, the chord between two points on the curve lies above the curve. Therefore, every trapezoid formed by the trapezoidal rule overestimates the actual area under x^2.
Hence the trapezoidal approximation is always greater than the exact integral 1/3.