Consider a non-negative function $f(x)$ which is continuous and bounded over the interval $[2,8]$. Let $M$ and $m$ denote, respectively, the maximum and the minimum values of $f(x)$ over the interval.
Among the combinations of $\alpha$ and $\beta$ given below, choose the one(s) for which the inequality
$$ \beta \leq \int_2^8 f(x) d x \leq \alpha $$
is guaranteed to hold.
Consider the Earth to be a perfect sphere of radius $R$. Then the surface area of the region, enclosed by the 60°N latitude circle, that contains the north pole in its interior is _______.
The value of the line integral $$\int_P^Q {({z^2}dx + 3{y^2}dy + 2xz\,dz)} $$ along the straight line joining the points $$P(1,1,2)$$ and $$Q(2,3,1)$$ is
The value of the integral $$\int\limits\!\!\!\int_R {xy\,dx\,dy} $$ over the regioin R, given in the figure, is _________ (rounded off to the nearest integer).
