81. The integral \[\int\limits_0^\pi {{{\sin }^3}\theta \,{\text{d}}\theta } \] is given by
82. 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 is
83. 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
84. 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
85. The area of the region bounded by the parabola y = x2 + 1 and the straight line x + y = 3 is
86. The maximum value of f(x) = x3 - 9x2 + 24x + 5 in the interval [1, 6] is
87. The value of the integral \[\int_0^{2\pi } {\left( {\frac{3}{{9 + {{\sin }^2}\theta }}} \right){\text{d}}\theta } \] is
88. 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
89. 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
90. At the point x = 0, the function f(x) = x3 has
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