Unit 5: Multivariable integration and applications - Subjective Questions

MTH165 — Mathematics For Engineers • Practice Questions with Detailed Answers

20 questions

Define the double integral of a function over a bounded region using Riemann sums. State the conditions under which the integral exists.

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

Evaluate $\iint_D(x+y)\,dA$ , where $D$ is the triangular region bounded by $x=0$ , $y=0$ , and $x+y=1$ .

Change the order of integration in $\int_0^1\int_{x^2}^{x}f(x,y)\,dy\,dx$ and use the result to find the area of the region.

Explain why a region may need to be split when setting up or changing the order of a double integral. Illustrate your answer for the triangle bounded by $y=0$ , $y=x$ , and $y=2-x$ .

Use a double integral to calculate the area enclosed by the curves $y=x$ and $y=x^2$ .

Use a double integral to find the volume below the paraboloid $z=4-x^2-y^2$ and above the $xy$ -plane.

Derive the Jacobian for the polar-coordinate transformation $x=r\cos\theta$ , $y=r\sin\theta$ , and state the resulting formula for a double integral.

Evaluate $\iint_D(x^2+y^2)\,dA,$ where $D$ is the disk $x^2+y^2\leq a^2$ .

Using a suitable change of variables, derive the area of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}\leq1.$

Define a triple integral and explain how Fubini's theorem is used to evaluate it.

Evaluate $\iiint_B(x+y+z)\,dV,$ where $B$ is the box $0\leq x\leq1$ , $0\leq y\leq2$ , and $0\leq z\leq3$ .

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

For the solid $D=\{(x,y,z):x\geq0,\ y\geq0,\ z\geq0,\ x+y+z\leq1\}$ , write the volume integral in three different orders and evaluate it.

Describe cylindrical coordinates, derive their volume element, and use a triple integral to obtain the volume of a right circular cylinder of radius $R$ and height $h$ .

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

Describe spherical coordinates and use them to evaluate the volume of a sphere of radius $a$ .

State and explain the change-of-variables theorem for double integrals. Why is the absolute value of the Jacobian required?

Use the transformation $u=x+y$ , $v=x-y$ to find the area of the region defined by $|x+y|\leq1$ and $|x-y|\leq1$ .

Compare the double-integral and triple-integral methods for finding volume, and calculate the volume enclosed between the paraboloids $z=x^2+y^2$ and $z=4-x^2-y^2$ .

Question 1

Define the double integral of a function over a bounded region using Riemann sums. State the conditions under which the integral exists.

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Answer 1

Definition: Let $D$ be a bounded region divided into small subregions $D_i$ having areas $\Delta A_i$ . If $(x_i^*,y_i^*)$ is a sample point in $D_i$ , then

$\iint_D f(x,y)\,dA=\lim_{n\to\infty}\sum_{i=1}^{n}f(x_i^*,y_i^*)\Delta A_i,$

provided the limit exists and is independent of the subdivision and sample points.

The largest diameter of the subregions must approach zero.
If $f$ is continuous on a closed and bounded region $D$ , then it is integrable over $D$ .
A bounded function with only finitely many discontinuities is also generally integrable when the boundary of $D$ is sufficiently regular.
Geometrically, when $f(x,y)\geq 0$ , the integral represents the volume below $z=f(x,y)$ and above $D$ .

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Question 5

Explain why a region may need to be split when setting up or changing the order of a double integral. Illustrate your answer for the triangle bounded by $y=0$ , $y=x$ , and $y=2-x$ .

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Answer 5

A region must be split when a single pair of boundary functions cannot describe all slices taken in the selected direction.

For vertical slices, the upper boundary changes at the intersection of $y=x$ and $y=2-x$ :

$x=2-x\implies x=1,\qquad y=1.$

Therefore, the integral must be split as

$\iint_D f(x,y)\,dA=\int_0^1\int_0^x f(x,y)\,dy\,dx+\int_1^2\int_0^{2-x}f(x,y)\,dy\,dx.$

For horizontal slices, one description covers the complete region:

$0\leq y\leq 1,\qquad y\leq x\leq 2-y.$

Hence, the reversed order is

$\iint_D f(x,y)\,dA=\int_0^1\int_y^{2-y}f(x,y)\,dx\,dy.$

This demonstrates that changing the order can replace a sum of integrals with a single integral.

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Question 18

State and explain the change-of-variables theorem for double integrals. Why is the absolute value of the Jacobian required?

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Answer 18

Let

$x=x(u,v),\qquad y=y(u,v)$

be a continuously differentiable, one-to-one transformation from a region $S$ in the $uv$ -plane to a region $D$ in the $xy$ -plane. If the Jacobian is nonzero in the interior of $S$ , then

$\iint_Df(x,y)\,dx\,dy=\iint_Sf(x(u,v),y(u,v))\left|\frac{\partial(x,y)}{\partial(u,v)}\right|du\,dv,$

where

$\frac{\partial(x,y)}{\partial(u,v)}=\begin{vmatrix}\frac{\partial x}{\partial u}&\frac{\partial x}{\partial v}\\\frac{\partial y}{\partial u}&\frac{\partial y}{\partial v}\end{vmatrix}.$

Explanation:

The Jacobian measures the local area-scaling factor of the transformation.
A small rectangle of area $du\,dv$ is transformed approximately into a parallelogram of area $|J|du\,dv$ .
A negative Jacobian indicates reversal of orientation, not negative area.
Therefore, the absolute value is required to ensure that transformed area elements remain nonnegative.

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Unit 5: Multivariable integration and applications - Subjective Questions

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