List of top Engineering Mathematics Questions on Integration

A stationary tank is cylindrical in shape with two hemispherical ends and is horizontal, as shown in the figure. \(R\) is the radius of the cylinder as well as of the hemispherical ends. The tank is half filled with an oil of density \(\rho\) and the rest of the space in the tank is occupied by air. The air pressure, inside the tank as well as outside it, is atmospheric. The acceleration due to gravity (\(g\)) acts vertically downward. The net horizontal force applied by the oil on the right hemispherical end (shown by the bold outline in the figure) is: 

If it is known that \( \int_0^{\infty} \frac{e^{ix}}{x} dx = \frac{\pi i}{2}, \) then compute: \( \int_0^{\infty} \frac{\sin 5x}{x} dx = ? \)

The area between the parabolas \( y^2 = 4 - x \) and \( y^2 = x \) is given by:
The value of the integral \[ \iiint\limits_{0}^{a, b, c} e^{x+y+z} \, dz \, dy \, dx \] is:
For applying Simpson's \( \frac{1}{3} \) rule, the given interval must be divided into how many number of sub-intervals?
The value of the integral \[ \iiint\limits_{0}^{a, b, c} e^{x+y+z} \, dz \, dy \, dx \] is:
The value of \[ I=\int_0^{\pi/2}\frac{(\sin x+\cos x)^2}{\sqrt{1+\sin 2x}}\,dx \] is:
The value of \[ \frac{4}{\pi} \int_0^{\pi/2} \sin^2 x \, dx \] is ________________ (rounded off to two decimal places).
Using divergence theorem, evaluate the integral \(\iint_{S} \vec{F} \cdot \vec{n}\, dA\), where \(S\) is the surface of the cone \(x^{2}+y^{2} \le z^{2},\ 0 \le z \le 3\). If \(\vec{F} = 4x\hat{i} + 3z\hat{j} + 5y\hat{k}\) is a vector function with outer unit normal vector \(\vec{n}\), the value of the integral is ______ (rounded off to the nearest integer).
Evaluation of the integral \[ \int \frac{dx}{\sqrt{2x - x^2}} \] results in
The integral \( \int \left( \frac{x}{2} - \frac{x^2}{3} + \frac{x^3}{4} - \cdots \right) \, dx \) is equal to
The value of the line integral \[ \oint_{C} (-3y\,dx + 3x\,dy + z\,dz) \] along the circle \( C: x^{2} + y^{2} = 1,\ z = 1 \) oriented in the clockwise sense as seen from the origin, is

The power consumption readings (in watt) by an instrument at fixed intervals of time (in seconds) are tabulated below:  

\[ \begin{array}{|c|c|} \hline \text{Time (s)} & \text{Power consumption (W)} \\ \hline 0.0 & 8.6 \\ 0.6 & 9.2 \\ 1.2 & 7.8 \\ 1.8 & 6.4 \\ 2.4 & 7.2 \\ 3.0 & 8.6 \\ 3.6 & 11.2 \\ \hline \end{array} \] 
Using Simpson's 1/3rd rule, the energy expenditure of the instrument in joules is _____.[Round off to two decimal places]

The value of \[ \lim_{x \to 0} \frac{1}{x} \int_{2}^{2+x} \left(t + \sqrt{t^2 + 5}\right) dt \]