List of top Mathematics Questions on Integral Calculus asked in IIT JAM MA

The double integral of \( f(x,y)=x \) over the triangular region with vertices at \( \left(-\frac{1}{2},\frac{1}{2}\right) \), \( (1,2) \), and \( (1,-1) \) is rounded off to one decimal place.
Let
The volume of the tetrahedron bounded by the planes \(x=1\), \(y=2\), \(z=3\) and \(12x+8y+6z=70\) is rounded off to one decimal place.
\[ \int_0^3 \left(|x-1|-x[x]\right)dx = \underline{} \] rounded off to one decimal place.
Let \( a = \lim_{n \to \infty} \left( \frac{1}{n^2} + \frac{2}{n^2} + \cdots + \frac{n-1}{n^2} \right) \) and \( b = \lim_{n \to \infty} \left( \frac{1}{n+1} + \frac{1}{n+2} + \cdots + \frac{1}{n+n} \right). \) Which of the following is/are true?

If \( a = \int_{\pi/3}^{\pi/6} \dfrac{\sin t + \cos t}{\sqrt{\sin 2t}} \, dt, \) then the value of \( \left( 2 \sin \frac{\alpha}{2} + 1 \right)^2 \text{ is ............}. \)