1

Evaluate the double integral $\iint_R x^2 y \, dx \, dy$ where $R$ is the rectangle defined by $0 \le x \le 1$ and $0 \le y \le 2$ .

2

Explain the concept of 'Change of Order of Integration' in double integrals and state why it is useful.

3

Change the order of integration for the integral $\int_0^1 \int_x^1 e^{y^2} \, dy \, dx$ and evaluate it.

4

Describe the process of changing variables in double integrals using the Jacobian.

5

Convert the integral $\iint_R \sqrt{x^2+y^2} \, dA$ to polar coordinates and evaluate, where $R$ is the region in the first quadrant bounded by the circle $x^2 + y^2 = a^2$ .

6

How can double integrals be applied to calculate the area of a plane region? Provide the general formula.

7

Find the area enclosed by the parabolas $y = x^2$ and $y = \sqrt{x}$ using a double integral.

8

Explain how double integrals are used to calculate the volume of a solid under a surface.

9

Set up and evaluate a triple integral to find the volume of a rectangular box bounded by the planes $x=0, x=a, y=0, y=b, z=0$ , and $z=c$ .

10

Evaluate the triple integral $\int_0^1 \int_0^x \int_0^{x+y} e^{x+y+z} \, dz \, dy \, dx$ .

11

Compare and contrast double and triple integrals in terms of their geometric interpretations and applications.

Comparison of Double and Triple Integrals:

Feature	Double Integrals ( $\iint$ )	Triple Integrals ( $\iiint$ )
Domain of Integration	A 2D region $R$ in the $xy$ -plane.	A 3D solid region $V$ in $xyz$ -space.
Differential Element	Area element $dA$ ( $dx\,dy$ or $r\,dr\,d\theta$ ).	Volume element $dV$ ( $dx\,dy\,dz$ , etc.).
Geometric Interpretation ( $f=1$ )	Represents the Area of the 2D region $R$ .	Represents the Volume of the 3D solid region $V$ .
Geometric Interpretation ( $f(x,y)$ or $f(x,y,z)$ )	Calculates the Volume under the surface $z=f(x,y)$ over region $R$ .	Calculates the "hypervolume" or physically represents total mass, charge, etc., if $f(x,y,z)$ is a density function.
Applications	- Area of plane regions.<br>- Volume under a surface.<br>- Mass, center of mass, moments of inertia for 2D laminas.	- Volume of 3D solids.<br>- Mass, center of mass, moments of inertia for 3D objects with variable density.
Coordinate Transformations	Often transformed to Polar coordinates using Jacobian $r$ .	Often transformed to Cylindrical (Jacobian $r$ ) or Spherical (Jacobian $\rho^2 \sin\phi$ ) coordinates.

12

Define the Jacobian used in change of variables for triple integrals from Cartesian to spherical coordinates, and state its value.

13

Use a triple integral to calculate the volume of a sphere of radius $R$ .

14

Change the order of integration for $\int_0^a \int_0^{\sqrt{a^2-x^2}} f(x,y) \, dy \, dx$ . Assume $a > 0$ .

15

Find the volume of the solid generated bounded by the cylinder $x^2 + y^2 = 4$ and the planes $z=0$ and $y+z=4$ .

16

What does a triple integral represent if the integrand is a density function $\rho(x,y,z)$ ? How does this differ from calculating purely geometric volume?

17

Describe the process of evaluating a triple integral using cylindrical coordinates.

18

Evaluate $\iint_R x y \, dA$ where $R$ is the region bounded by $x=0$ , $y=0$ , and $x+y=1$ .

19

Change the variables to evaluate the double integral $\iint_R (x+y)^2 \, dx \, dy$ where $R$ is the parallelogram bounded by the lines $x+y=0, x+y=2, 3x-2y=0$ , and $3x-2y=3$ .

20

Set up the triple integral to find the volume of a right circular cone of base radius $R$ and height $h$ using cylindrical coordinates.

Unit6 - Subjective Questions