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

Let \( C \) be the boundary of the square with vertices \( (0,0), (1,0), (1,1), (0,1) \) oriented counterclockwise. Then the value of the line integral \[ \oint_C x^2y^2 dx + (x^2 - y^2) dy \] is ............ (rounded off to two decimal places).

If \( \displaystyle \int_0^1 \int_{2y}^2 e^{x^2} \, dx \, dy = k(e^4 - 1), \) then \( k \) equals ...............

Let \( S \) be that part of the surface of the paraboloid \( z = 16 - x^2 - y^2 \) which is above the plane \( z = 0 \) and \( D \) be its projection on the xy-plane. Then the area of \( S \) equals

The value of the triple integral \( \iiint_V (x^2y + 1) \, dx\,dy\,dz \), where \( V \) is the region given by \( x^2 + y^2 \le 1, \, 0 \le z \le 2, \) is

The area bounded by the curves \( x^2 + y^2 = 2x \) and \( x^2 + y^2 = 4x \), and the straight lines \( y = x \) and \( y = 0 \) is

Let \( S \) be the surface of the portion of the sphere with centre at the origin and radius 4, above the \( xy \)-plane. Let \( \vec{F} = y\hat{i} - x\hat{j} + yxz^3\hat{k}. \) If \( \hat{n} \) is the unit outward normal to \( S \), then
\[ \iint_S (\nabla \times \vec{F}) \cdot \hat{n} \, dS \] equals

Let \( \vec{a} = \hat{i} + \hat{j} + \hat{k} \) and \( \vec{r} = x\hat{i} + y\hat{j} + z\hat{k}, \, x, y, z \in \mathbb{R}. \) Which of the following is FALSE?

Let \( S \) be the part of the cone \( z^2 = x^2 + y^2 \) between the planes \( z = 0 \) and \( z = 1. \) Then the value of the surface integral \( \iint_S (x^2 + y^2) \, dS \) is

Let \( \vec{F} = x\hat{i} + y\hat{j} + z\hat{k} \) and \( S \) be the sphere given by \( (x - 2)^2 + (y - 2)^2 + (z - 2)^2 = 4. \) If \( \hat{n} \) is the unit outward normal to \( S \), then \( \dfrac{1}{\pi} \iint_S \vec{F} \cdot \hat{n} \, dS \) is .............

If \( \hat{F}(x, y) = (3x - 8y) \hat{i} + (4y - 6xy) \hat{j} \) for \( (x, y) \in \mathbb{R}^2 \), then \( \oint_C \vec{F} \cdot d \vec{r} \), where \( C \) is the boundary of the triangular region bounded by the lines \( x = 0 \), \( y = 0 \), and \( x + y = 1 \) oriented in the anti-clockwise direction, is

Consider the region \( D \) in the yz-plane bounded by the line \( y = \frac{1}{2} \) and the curve \( y^2 + z^2 = 1 \), where \( y \geq 0 \). If the region \( D \) is revolved about the \( z \)-axis in \( \mathbb{R}^3 \), then the volume of the resulting solid is

If the volume of the solid in \( \mathbb{R}^3 \) bounded by the surfaces \[ x = -1, \ x = 1, \ y = -1, \ y = 1, \ z = 2, \ y^2 + z^2 = 2 \] is \( \alpha - \pi \), then \( \alpha = \text{.........}. \)

The value of the integral \[ \int_0^1 \int_x^1 y^4 e^{x y^2} \, dy \, dx\] is .........… (correct up to three decimal places).

The area of the parametrized surface \[ S = \left\{ \left( (2 + \cos u) \cos v, (2 + \cos u) \sin v, \sin u \right) \in \mathbb{R}^3 \mid 0 \leq u \leq \frac{\pi}{2}, 0 \leq v \leq \frac{\pi}{2} \right\} \] is .................. (correct up to two decimal places).