Calculus MCQs Question and Answer in Engineering Maths

The integral \[\int\limits_0^\pi {{{\sin }^3}\theta \,{\text{d}}\theta } \]   is given by

The volume of an object expressed in spherical co-ordinates is given by \[{\text{V}} = \int_0^{2\pi } {\int_0^{\frac{\pi }{3}} {\int_0^1 {{{\text{r}}^2}\sin \phi \,{\text{dr d}}\phi \,{\text{d}}\theta .} } } \]
The value of the integral is

The value of the integral \[\int_0^\infty {\int_0^\infty {{{\text{e}}^{ - {{\text{x}}^2}}}{{\text{e}}^{ - {{\text{y}}^2}}}} } {\text{dx dy}}\]     is

The range of value of k for which the function f(x) = (k² - 4)x² + 6x³ + 8x⁴ has a local maxima at point x = 0 is

The area of the region bounded by the parabola y = x² + 1 and the straight line x + y = 3 is

The maximum value of f(x) = x³ - 9x² + 24x + 5 in the interval [1, 6] is

The value of the integral \[\int_0^{2\pi } {\left( {\frac{3}{{9 + {{\sin }^2}\theta }}} \right){\text{d}}\theta } \]     is

A velocity vector is given as \[\overrightarrow {\text{V}} = 5{\text{xy}}\overrightarrow {\text{i}} + 2{{\text{y}}^2}\overrightarrow {\text{j}} + 3{\text{y}}{{\text{z}}^2}\overrightarrow {\text{k}} .\]       The divergence of this velocity vector at (1, 1, 1) is

Given a vector field \[\overrightarrow {\rm{F}} = {{\rm{y}}^2}{\rm{x}}{{{\rm{\hat a}}}_{\rm{x}}} - {\rm{yz}}{{{\rm{\hat a}}}_{\rm{y}}} - {{\rm{x}}^2}{{{\rm{\hat a}}}_{\rm{z}}},\]     the line integral \[\int {\overrightarrow {\text{F}} \cdot \overrightarrow {{\text{d}}l} } \]  evaluated along a segment on the x-axis from x = 1 to x = 2 is

At the point x = 0, the function f(x) = x³ has