Question

The integral  \(\mathop \smallint \limits_{{x_1}}^{{x_2}} {x^2}dx\) with  \({x_2} > {x_1} > 0\)  is evaluated analytically as well as numerically using a single application of the trapezoidal rule. If I is the exact value of the integral obtained analytically and J is the approximate value obtained using the trapezoidal rule, which of the following statements is correct about their relationship?

1. J > I
2. J < I
3. J = I
4. Insufficient data to determine the relationship

Answer 1

Correct Answer - Option 1 : J > I

Exact value of integration is computed by integration which follows the exact shape of graph while computing the area.

Whereas, in Trapezoidal rule, the lines joining each points are considered straight line which in not the exact variation of graph all the time

Also we know that approximate value calculated by trapezoidal rule is always greater than the exact value calculated by integration

∴ J > I

Where, J = approximate value, I = Exact value