61. The value of the integral of the function g(x, y) = 4x3 + 10y4 along the straight line segment from the point (0, 0) to the point (1, 2) in the x - y plane is
62. The value of \[\mathop {\lim }\limits_{{\rm{x}} \to 0} \frac{{1 - \cos \left( {{{\rm{x}}^2}} \right)}}{{2{x^4}}}\] is
63. Consider points P and Q in the x-y plane, with P = (1, 0) and Q = (0, 1). The line integral
\[2\int\limits_{\rm{P}}^{\rm{Q}} {\left( {{\rm{xdx}} + {\rm{ydy}}} \right)} \] along the semicircle with the line segment PQ as its diameter
64. A surface S(x, y) = 2x + 5y - 3 is integrated once over a path consisting of the points that satisfy (x + 1)2 + (y - 1)2 = √2. The integral evaluates to
65. Let w = f(x, y), where x and y are functions of t. Then, according to the chain rule, \[\frac{{{\rm{dw}}}}{{{\rm{dt}}}}\] is equal
66. For the parallelogram OPQR shown in the sketch, \[\overline {{\rm{OP}}} = {\rm{a\hat t}} + {\rm{b\hat j}}\] and \[\overline {{\rm{OR}}} = {\rm{c\hat t}} + {\rm{d\hat j}}{\rm{.}}\] The area of the parallelogram is
67. The distance between the origin and the point nearest to it on the surface z2 = 1 + xy is
68. The value of \[\int\limits_0^3 {\int\limits_0^{\rm{x}} {\left( {6 - {\rm{x}} - {\rm{y}}} \right)} {\rm{dx\,dy}}} \] is
69. If for non-zero x, \[{\rm{af}}\left( {\rm{x}} \right) + {\rm{bf}}\left( {\frac{1}{{\rm{x}}}} \right) = \frac{1}{{\rm{x}}} - 25\] where a ≠ b then \[\int\limits_1^2 {{\rm{f}}\left( {\rm{x}} \right){\rm{dx}}} \] is
70. The improper integral \[\int\limits_0^\infty {{{\rm{e}}^{ - 2{\rm{t}}}}} {\rm{dt}}\] converges to
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