Unit 5: Multivariable integration and applications - Subjective Questions

MTH165 — Mathematics For Engineers • Practice Questions with Detailed Answers

20 questions

Define a double integral over a rectangular region and explain its geometric interpretation.

Evaluate the double integral $\displaystyle \int_0^1\int_0^2(x+2y)\,dy\,dx$ .

Explain the difference between Type I and Type II plane regions used in double integration.

Change the order of integration in $\displaystyle \int_0^1\int_x^1 f(x,y)\,dy\,dx$ and describe the region.

Change the order of integration and evaluate $\displaystyle I=\int_0^1\int_x^1 e^{y^2}\,dy\,dx$ .

Evaluate $\displaystyle \iint_R xy\,dA$ , where $R$ is bounded by $y=x^2$ and $y=x$ in the first quadrant.

State Fubini's theorem for double integrals and explain its significance.

Define a triple integral and explain its physical and geometric interpretations.

Evaluate $\displaystyle \iiint_V(x+y+z)\,dV$ , where $V=[0,1]\times[0,2]\times[0,3]$ .

Describe Cartesian, cylindrical, and spherical coordinate systems, including their volume elements.

Explain the change-of-variables formula for double integrals and define the Jacobian.

Derive the area element in polar coordinates using the Jacobian.

Use polar coordinates to evaluate $\displaystyle \iint_D(x^2+y^2)\,dA$ , where $D$ is the disk $x^2+y^2\leq a^2$ .

Using a double integral, find the area enclosed between the curves $y=x$ and $y=x^2$ .

Use a double integral to find the volume below the plane $z=6-x-2y$ and above the triangular region in the $xy$ -plane bounded by $x=0$ , $y=0$ , and $x+2y=6$ .

Find the volume under the paraboloid $z=4-x^2-y^2$ and above the $xy$ -plane.

Use a triple integral in cylindrical coordinates to derive the volume of a right circular cylinder of radius $a$ and height $h$ .

Use a triple integral in spherical coordinates to derive the volume of a sphere of radius $a$ .

Find the volume of the tetrahedron in the first octant bounded by the coordinate planes and the plane $\displaystyle \frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$ using a triple integral.

Apply the transformation $u=x+y$ and $v=x-y$ to evaluate $\displaystyle \iint_R(x+y)^2\,dA$ , where $R$ is bounded by $x+y=0$ , $x+y=2$ , $x-y=0$ , and $x-y=1$ .

Question 1

Define a double integral over a rectangular region and explain its geometric interpretation.

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Answer 1

Definition: Let $f(x,y)$ be continuous on the rectangle $R=[a,b]\times[c,d]$ . Its double integral is defined as the limit of Riemann sums:

$\iint_R f(x,y)\,dA=\lim_{m,n\to\infty}\sum_{i=1}^{m}\sum_{j=1}^{n}f(x_i^*,y_j^*)\,\Delta x\,\Delta y.$

Here, $\Delta x=(b-a)/m$ and $\Delta y=(d-c)/n$ .

If $f(x,y)\geq 0$ , the integral represents the volume below the surface $z=f(x,y)$ and above $R$ .
If $f$ assumes both positive and negative values, it gives the signed volume.
By Fubini's theorem, for continuous $f$ ,

$\iint_R f(x,y)\,dA=\int_a^b\int_c^d f(x,y)\,dy\,dx=\int_c^d\int_a^b f(x,y)\,dx\,dy.$

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Question 7

State Fubini's theorem for double integrals and explain its significance.

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Answer 7

Fubini's theorem: If $f(x,y)$ is continuous on the rectangle $R=[a,b]\times[c,d]$ , then

$\iint_R f(x,y)\,dA=\int_a^b\left(\int_c^d f(x,y)\,dy\right)dx$

$=\int_c^d\left(\int_a^b f(x,y)\,dx\right)dy.$

Significance:

It converts a double integral into two successive single integrals.
It permits the order of integration to be interchanged under suitable conditions.
It allows the easier order to be selected when one iterated integral is difficult.
For nonrectangular regions, the limits must be adjusted according to the geometry of the region.
A more general version applies to integrable functions, even when continuity is not available everywhere.

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Unit4 Unit6

Unit 5: Multivariable integration and applications - Subjective Questions

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