A scalar potential $$\,\,\varphi \,\,$$ has the following gradient: $$\,\,\nabla \varphi = yz\widehat i + xz\widehat j + xy\widehat k.\,\,$$ Consider the integral $$\,\,\int_C {\nabla \varphi .d\overrightarrow r \,\,} $$ on the curve $$\overrightarrow r = x\widehat i + y\widehat j + z\widehat k.\,\,$$ The curve $$C$$ is parameterized as follows: $$\,\,\left\{ {\matrix{ {x = t} \cr {y = {t^2}} \cr {z = 3{t^2}} \cr } \,\,\,\,\,\,\,} \right.$$ and $$1 \le t \le 3.\,\,\,\,\,$$
The value of the integral is _________.

The probability that a screw manufactured by a company is defective is $$0.1$$. The company sells screws in packets containing $$5$$ screws and gives a guarantee of replacement if one or more screws in the packet are found to be defective. The probability that a packet would have to be replaced is _________.

Numerical integration using trapezoidal rule gives the best result for a single variable function, which is

The error in numerically computing the integral $$\,\int\limits_0^\pi {\left( {\sin \,x + \cos \,x} \right)dx\,\,\,} $$ using the trapezoidal rule with three intervals of equal length between $$0$$ and $$\pi $$ is _________.