Some JEE problems ask for bounds on an integral rather than exact values. If m <= f(x) <= M on [a, b], then m(b - a) <= integral from a to b of f(x) dx <= M(b - a). Tighter bounds use the monotonicity of f: if f is increasing on [a, b], then f(a)(b - a) <= integral <= f(b)(b - a). For example, to bound the integral from 1 to 2 of sqrt(1 + $x^3$) dx: on [1, 2], 1 + $x^3$ ranges from 2 to 9, so sqrt(2) <= sqrt(1 + $x^3$) <= 3. Hence sqrt(2) <= integral <= 3. This technique is tested in integer-answer type questions where you need to identify which integer the integral is closest to.