List of top Mechanical Engineering Questions on Calculus

Match List-I with List-II.

List-I (Complex integrate)List-II (Value)
A. \( \displaystyle \int_{0}^{2\pi} \cos^2 x \, dx \)
B. \( \displaystyle \int_{0}^{\pi} \cos x \, dx \)
C. \( \displaystyle \oint_C dz \), where \(C\) is a simple closed curve
D. \( \displaystyle \oint_C \frac{dz}{z^2-1} \), around circle \(|z-1|=1\)		I. \(4\pi i\)
II. \(0\)
III. \(\frac{\pi}{4}\)
IV. \( \frac{1}{4}\cosh 2 - \frac{1}{2}\sinh 2 + \frac{1}{2}\pi i \sinh 2 \)

Match List-I with List-II.

List-I (Cauchy Integral)List-II (Value)
A. \( \displaystyle \oint_C \frac{\sin z + \cos \pi z}{(z-1)(z-2)} dz \)
B. \( \displaystyle \oint_C \frac{dz}{(z+1)^4} \)
C. \( \displaystyle \oint_C \frac{e^z}{(z+2)^2} dz \)
D. \( \displaystyle \oint_C \frac{\sin z}{z^2} dz \)		I. \( \frac{8\pi i e^{-2}}{3} \)
II. \(4\pi i\)
III. \( \frac{\pi i}{32} \)
IV. \( \frac{i}{\pi} \)

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The value of \(\Gamma\left(\frac{1}{2}\right)\) is:
The volume of the solid bounded by the planes \(x=0\), \(y=0\), \(z=0\) and \(x+y+z=2\) is:
What are the absolute maximum value and the absolute minimum value of a function \( f(x) = \sin x + \cos x \) in the interval \( [0, \pi] \)?
If \( y = e^{(x+e)^{(x+e)^{(x+\cdots)}}} \), what is the value of \( \frac{d}{dx}(y) \)?
The value of the integral \( \int_C \frac{3\sigma^2 + x}{z^2 - 1} \, dz \), where \( C \) is the circle \( |z - 1| = 1 \), is
The value of the integral
\[ \oint_C \left( y^3 \mathbf{i} - x^3 \mathbf{j} \right) \cdot \left( i \, dx + j \, dy \right) \] where \(C\) is the closed curve, is: