Partial Differentiation: Part 3

Advanced Numerical Problems on Partial Differentiation

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Problem 1: Laplace’s Equation

Problem: Verify that \( u = x^3 - 3xy^2 \) satisfies Laplace’s equation.

Show / Hide Solution

\[ u_x = 3x^2 - 3y^2, \quad u_y = -6xy \] \[ u_{xx} = 6x, \quad u_{yy} = -6x \] \[ u_{xx} + u_{yy} = 6x - 6x = 0 \]

Hence verified.

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Problem 2: Total Differential

Problem: If \( z = e^{xy}\sin(x+y) \), find total differential \( dz \).

Show / Hide Solution

\[ dz = z_x dx + z_y dy \] \[ z_x = e^{xy}[y\sin(x+y)+\cos(x+y)] \] \[ z_y = e^{xy}[x\sin(x+y)+\cos(x+y)] \] \[ dz = e^{xy}[y\sin(x+y)+\cos(x+y)]dx \] \[ + e^{xy}[x\sin(x+y)+\cos(x+y)]dy \]

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Problem 3: Directional Derivative

Problem: Find directional derivative of \( \phi=x^2y+yz^3 \) at (1,-1,1) in direction \(2\mathbf{i}-\mathbf{j}+2\mathbf{k}\).

Show / Hide Solution

\[ \nabla \phi = (2xy)\mathbf{i} + (x^2+z^3)\mathbf{j} + (3yz^2)\mathbf{k} \] At (1,-1,1): \[ \nabla \phi = (-2)\mathbf{i} + 2\mathbf{j} - 3\mathbf{k} \] Direction vector magnitude: \[ |\vec{a}| = 3 \] Unit vector: \[ \hat{a}=\frac{1}{3}(2,-1,2) \] Directional derivative: \[ \nabla\phi \cdot \hat{a} = -4 \]

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Problem 4: Taylor Expansion

Problem: Expand \( f(x,y)=e^x\cos y \) about (0,0) up to second order.

Show / Hide Solution

\[ f(0,0)=1 \] \[ f_x(0,0)=1, \quad f_y(0,0)=0 \] \[ f_{xx}=1,\quad f_{yy}=-1,\quad f_{xy}=0 \] \[ f(x,y)=1+x+\frac{1}{2}(x^2-y^2) \]

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Problem 5: Jacobian

Problem: If \( x=r\cos\theta, y=r\sin\theta \), find Jacobian \( \frac{\partial(x,y)}{\partial(r,\theta)} \).

Show / Hide Solution

\[ J= \begin{vmatrix} \cos\theta & -r\sin\theta \\ \sin\theta & r\cos\theta \end{vmatrix} \] \[ = r(\cos^2\theta+\sin^2\theta)=r \]

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Problem 6: Lagrange Multiplier

Problem: Find extrema of \( f=x^2+y^2 \) subject to \( x+y=1 \).

Show / Hide Solution

\[ \nabla f=(2x,2y), \quad \nabla g=(1,1) \] \[ (2x,2y)=\lambda(1,1) \Rightarrow x=y \] Using constraint: \[ 2x=1 \Rightarrow x=y=\frac12 \] Minimum value: \[ f=\frac12 \]

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Problem 7: Implicit Differentiation

Problem: If \( x^2+y^2+xy=7 \), find \( \frac{dy}{dx} \).

Show / Hide Solution

\[ 2x+2y\frac{dy}{dx}+x\frac{dy}{dx}+y=0 \] \[ (2y+x)\frac{dy}{dx}=-(2x+y) \] \[ \frac{dy}{dx}=-\frac{2x+y}{2y+x} \]

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Problem 8: Harmonic Function

Problem: Show \( u=\ln(x^2+y^2) \) is harmonic.

Show / Hide Solution

\[ u_x=\frac{2x}{x^2+y^2} \] \[ u_{xx}=\frac{2(y^2-x^2)}{(x^2+y^2)^2} \] \[ u_{yy}=\frac{2(x^2-y^2)}{(x^2+y^2)^2} \] \[ u_{xx}+u_{yy}=0 \] Hence harmonic.

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Problem 9: Euler’s Theorem

Problem: Verify Euler’s theorem for \( u=(x^2+y^2)^{3/2} \).

Show / Hide Solution

\[ u_x=3x(x^2+y^2)^{1/2} \] \[ u_y=3y(x^2+y^2)^{1/2} \] \[ xu_x+yu_y=3(x^2+y^2)^{3/2}=3u \]

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Problem 10: Maximum Directional Derivative

Problem: Find maximum rate of increase of \( f=xye^z \) at (1,1,0).

Show / Hide Solution

\[ \nabla f= (ye^z)\mathbf{i} + (xe^z)\mathbf{j} + (xye^z)\mathbf{k} \] At (1,1,0): \[ \nabla f=(1,1,1) \] Maximum rate: \[ |\nabla f|=\sqrt3 \] Direction: along gradient vector.

Partial Differentiation: Part 2

Numerical Examples on Partial Differentiation

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Example 1: First Order Partial Derivatives

Problem: If \( z = 3x^2y + 2xy^3 - 5y \), find \( \frac{\partial z}{\partial x} \) and \( \frac{\partial z}{\partial y} \).

Show / Hide Solution

Differentiate w.r.t \(x\):

\[ \frac{\partial z}{\partial x} = 6xy + 2y^3 \]

Differentiate w.r.t \(y\):

\[ \frac{\partial z}{\partial y} = 3x^2 + 6xy^2 - 5 \]

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Example 2: Second Order Partial Derivatives

Problem: If \( z = x^3y^2 + 4xy \), find \( \frac{\partial^2 z}{\partial x^2} \).

Show / Hide Solution

First derivative:

\[ \frac{\partial z}{\partial x} = 3x^2y^2 + 4y \]

Second derivative:

\[ \frac{\partial^2 z}{\partial x^2} = 6xy^2 \]

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Example 3: Mixed Partial Derivatives

Problem: If \( z = x^2y^3 \), verify mixed partial derivatives.

Show / Hide Solution

\[ \frac{\partial z}{\partial x} = 2xy^3 \] \[ \frac{\partial^2 z}{\partial y \partial x} = 6xy^2 \] \[ \frac{\partial z}{\partial y} = 3x^2y^2 \] \[ \frac{\partial^2 z}{\partial x \partial y} = 6xy^2 \]

Hence verified.

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Example 4: Chain Rule

Problem: If \( z = x^2 + y^2 \), where \( x = r\cos\theta \), \( y = r\sin\theta \), find \( \frac{\partial z}{\partial r} \).

Show / Hide Solution

\[ z = r^2(\cos^2\theta + \sin^2\theta) = r^2 \] \[ \frac{\partial z}{\partial r} = 2r \]

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Example 5: Implicit Differentiation

Problem: If \( x^2 + y^2 + z^2 = 1 \), find \( \frac{\partial z}{\partial x} \).

Show / Hide Solution

\[ 2x + 2z\frac{\partial z}{\partial x} = 0 \] \[ \frac{\partial z}{\partial x} = -\frac{x}{z} \]

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Example 6: Euler’s Theorem

Problem: Verify Euler’s theorem for \( z = x^2y^3 \).

Show / Hide Solution

\[ \frac{\partial z}{\partial x} = 2xy^3 \] \[ \frac{\partial z}{\partial y} = 3x^2y^2 \] \[ x\frac{\partial z}{\partial x} + y\frac{\partial z}{\partial y} = 5x^2y^3 = 5z \]

Hence verified.

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Example 7: Total Derivative

Problem: If \( z = x^2y + y^3 \), where \( x=t^2 \), \( y=t \), find \( \frac{dz}{dt} \).

Show / Hide Solution

\[ \frac{dz}{dt} = \frac{\partial z}{\partial x}\frac{dx}{dt} + \frac{\partial z}{\partial y}\frac{dy}{dt} \] \[ = (2xy)(2t) + (x^2+3y^2) \] Substitute \(x=t^2, y=t\): \[ \frac{dz}{dt} = 5t^4 + 3t^2 \]

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Example 8: Tangent Plane

Problem: Find tangent plane to \( z=x^2+y^2 \) at (1,1,2).

Show / Hide Solution

\[ f_x=2x,\quad f_y=2y \] \[ z-2=2(x-1)+2(y-1) \] \[ z=2x+2y-2 \]

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Example 9: Maxima and Minima

Problem: Find stationary points of \( z=x^2+y^2-4x-6y \).

Show / Hide Solution

\[ 2x-4=0 \Rightarrow x=2 \] \[ 2y-6=0 \Rightarrow y=3 \]

Minimum at (2,3).

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Example 10: Jacobian

Problem: If \( u=x+y \), \( v=x-y \), find \( \frac{\partial(u,v)}{\partial(x,y)} \).

Show / Hide Solution

\[ J= \begin{vmatrix} 1 & 1 \\ 1 & -1 \end{vmatrix} =-2 \]

Application of Double Integration 1

Applications of Double Integrals

Double integration is a fundamental tool in multivariable calculus used to compute volumes, areas, mass, centroids, and physical quantities over two-dimensional regions.

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1. Volume Under a Surface

If $z = f(x,y)$ over region $R$, the volume is:

$$ V = \iint_R f(x,y)\, dA $$

Illustration

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2. Area of a Plane Region

The area of region $R$:

$$ A = \iint_R 1\, dA $$

Example Region (Circle)

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3. Mass of a Lamina

If density is $\rho(x,y)$:

$$ M = \iint_R \rho(x,y)\, dA $$

If density is constant $\rho = k$:

$$ M = k \cdot \text{Area}(R) $$

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4. Center of Mass (Centroid)

$$ \bar{x} = \frac{1}{M} \iint_R x \rho(x,y)\, dA $$ $$ \bar{y} = \frac{1}{M} \iint_R y \rho(x,y)\, dA $$

Centroid Illustration

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5. Moments of Inertia

$$ I_x = \iint_R y^2 \rho(x,y)\, dA $$ $$ I_y = \iint_R x^2 \rho(x,y)\, dA $$ $$ I_0 = \iint_R (x^2 + y^2)\rho(x,y)\, dA $$

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6. Probability Applications

If $f(x,y)$ is a joint probability density function:

$$ P((x,y)\in R) = \iint_R f(x,y)\, dA $$ $$ \iint_{\mathbb{R}^2} f(x,y)\, dA = 1 $$

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7. Surface Area

If $z = f(x,y)$:

$$ A = \iint_R \sqrt{1 + \left(\frac{\partial f}{\partial x}\right)^2 + \left(\frac{\partial f}{\partial y}\right)^2} \, dA $$

Surface Illustration

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End of Notes

Applications of Double Integrals

Area Enclosed by Plane Curves

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Illustration 16.1

Find the area of a quadrant of the circle

$$x^2 + y^2 = a^2$$

Solution:

$$ A=\int_{0}^{a}\int_{0}^{\sqrt{a^2-x^2}} dy\,dx = \int_{0}^{a} \sqrt{a^2-x^2}\, dx $$

$$ A=\frac{\pi a^2}{4} $$

Answer: $$\boxed{\frac{\pi a^2}{4}}$$

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Illustration 16.2

Find the area bounded by

$$y = 2-x, \qquad y^2 = 2(x+2)$$

Intersection:

$$ (2-x)^2=2(x+2) $$ $$ x^2-6x=0 \Rightarrow x=0,6 $$

Using horizontal strips: $$ A=\int_{-4}^{2}\left(4-y-\frac{y^2}{2}\right)dy $$

$$ A=18 $$

Answer: $$\boxed{18}$$

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Illustration 16.3

Find the area bounded by

$$2x-3y+4=0,\quad x+y-3=0,\quad y=0$$

Intersection Points:

$$(-2,0),\; (3,0),\; (1,2)$$

$$ A=\int_{0}^{2}\int_{\frac{3y-4}{2}}^{3-y} dx\,dy $$

$$ A=5 $$

Answer: $$\boxed{5}$$

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Illustration 16.4

Find the area bounded by

$$x^2=4(y+2), \quad x^2=3-y$$

Intersection:

$$y=-1, \quad x=\pm 2$$

$$ A=\int_{-2}^{2}\left(5-\frac{5x^2}{4}\right)dx $$

$$ A=\frac{172}{15} $$

Answer: $$\boxed{\frac{172}{15}}$$

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Illustration 16.5

Find the area bounded by

$$x(x^2+y^2)=a(x^2-y^2)$$

Using polar substitution: $$x=a\cos\theta$$

$$ A=4a^2\int_{0}^{\pi/2} \left(\sin^2\frac{\theta}{2} -2\sin^4\frac{\theta}{2}\right)d\theta $$

$$ A=\frac{\pi a^2}{2} $$

Answer: $$\boxed{\frac{\pi a^2}{2}}$$

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End of Chapter Section

Application of Double Integration

Applications of Double Integrals

Area Enclosed by Plane Curves

Double integration is used to compute area, volume, mass and many physical quantities.

$$A = \iint_R 1\, dA$$

Example 1

Find the area of a quadrant of $x^2 + y^2 = a^2$.

🔽 Show Step-by-Step Solution

Using vertical strips: $$A=\int_0^a \int_0^{\sqrt{a^2-x^2}} dy\,dx$$

$$A=\int_0^a \sqrt{a^2-x^2}\,dx$$

Using standard integral: $$\int \sqrt{a^2-x^2}dx =\frac{x}{2}\sqrt{a^2-x^2} +\frac{a^2}{2}\sin^{-1}\left(\frac{x}{a}\right)$$

Evaluating limits: $$A=\frac{\pi a^2}{4}$$

Final Answer: $$\boxed{\frac{\pi a^2}{4}}$$

Example 2

Find the area bounded by $y=2-x$, $y^2=2(x+2)$.

🔽 Show Step-by-Step Solution

Intersection points: $$(2-x)^2=2(x+2)$$ $$x=0,6$$

Using horizontal strips: $$A=\int_{-4}^{2}\int_{(y^2/2)-2}^{2-y} dx\,dy$$

$$A=\int_{-4}^{2}\left(4-y-\frac{y^2}{2}\right) dy$$

After evaluation: $$A=18$$

Final Answer: $$\boxed{18}$$

Example 3

Find the area bounded by $2x-3y+4=0$, $x+y-3=0$, $y=0$.

🔽 Show Step-by-Step Solution

Vertices: $$(-2,0), (3,0), (1,2)$$

$$A=\int_{0}^{2}\int_{\frac{3y-4}{2}}^{3-y} dx\,dy$$

$$A=\int_{0}^{2}\left(5-\frac{5y}{2}\right) dy$$

$$A=5$$

Final Answer: $$\boxed{5}$$

Example 4

Find the area bounded by $x^2=4(y+2)$ and $x^2=3-y$.

🔽 Show Step-by-Step Solution

Intersection: $$4(y+2)=3-y$$ $$y=-1,\quad x=\pm2$$

$$A=\int_{-2}^{2}\int_{\frac{x^2}{4}-2}^{3-x^2} dy\,dx$$

$$A=\int_{-2}^{2}\left(5-\frac{5x^2}{4}\right) dx$$

$$A=\frac{172}{15}$$

Final Answer: $$\boxed{\frac{172}{15}}$$

Example 5

Find the area bounded by $x(x^2+y^2)=a(x^2-y^2)$.

🔽 Show Step-by-Step Solution

Using polar substitution: $$r = a\cos(2\theta)\sec\theta$$

Area of full loop: $$A=4a^2\int_{0}^{\pi/2} \left(\sin^2\frac{\theta}{2} -2\sin^4\frac{\theta}{2}\right)d\theta$$

After evaluation: $$A=\frac{\pi a^2}{2}$$

Final Answer: $$\boxed{\frac{\pi a^2}{2}}$$

Conclusion

Double integrals are powerful tools for computing area, volume, and many physical applications.

LDLT Factorization

LDL^T Factorization – Detailed Numerical Examples

Dr. Brajesh Kumar Jha

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General Formula

For symmetric matrix $A$:

$$ A = L D L^T $$ $$ d_k = a_{kk} - \sum_{j=1}^{k-1} l_{kj}^2 d_j $$ $$ l_{ik} = \frac{1}{d_k} \left( a_{ik} - \sum_{j=1}^{k-1} l_{ij} l_{kj} d_j \right) $$

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Example 1

$$ A = \begin{bmatrix} 4 & 2 & 2 \\ 2 & 5 & 1 \\ 2 & 1 & 3 \end{bmatrix} $$

Step 1

$$ d_1 = 4 $$ $$ l_{21} = \frac{2}{4} = \frac{1}{2}, \quad l_{31} = \frac{2}{4} = \frac{1}{2} $$

Step 2

$$ d_2 = 5 - \left(\frac{1}{2}\right)^2(4) $$ $$ = 5 - 1 = 4 $$ $$ l_{32} = \frac{1}{4} \left( 1 - \frac{1}{2}\cdot\frac{1}{2}\cdot 4 \right) = \frac{1}{4}(1-1)=0 $$

Step 3

$$ d_3 = 3 - \left(\frac{1}{2}\right)^2(4) $$ $$ = 3 - 1 = 2 $$

Final Matrices

$$ L = \begin{bmatrix} 1 & 0 & 0 \\ \frac{1}{2} & 1 & 0 \\ \frac{1}{2} & 0 & 1 \end{bmatrix} $$ $$ D = \begin{bmatrix} 4 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 2 \end{bmatrix} $$

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Example 2

$$ A = \begin{bmatrix} 2 & -1 & 1 \\ -1 & 2 & -1 \\ 1 & -1 & 2 \end{bmatrix} $$

Step 1

$$ d_1 = 2 $$ $$ l_{21} = -\frac{1}{2}, \quad l_{31} = \frac{1}{2} $$

Step 2

$$ d_2 = 2 - \left(-\frac{1}{2}\right)^2(2) $$ $$ = 2 - \frac{1}{2} = \frac{3}{2} $$ $$ l_{32} = \frac{1}{\frac{3}{2}} \left( -1 - \left(\frac{1}{2}\right)\left(-\frac{1}{2}\right)(2) \right) $$ $$ = \frac{2}{3}\left(-1+\frac{1}{2}\right) = -\frac{1}{3} $$

Step 3

$$ d_3 = 2 - \left(\frac{1}{2}\right)^2(2) - \left(-\frac{1}{3}\right)^2\left(\frac{3}{2}\right) $$ $$ = 2 - \frac{1}{2} - \frac{1}{6} = \frac{4}{3} $$

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Example 3

$$ A = \begin{bmatrix} 9 & 3 & 6 \\ 3 & 5 & 4 \\ 6 & 4 & 10 \end{bmatrix} $$

Step 1

$$ d_1 = 9 $$ $$ l_{21} = \frac{3}{9}=\frac{1}{3}, \quad l_{31} = \frac{6}{9}=\frac{2}{3} $$

Step 2

$$ d_2 = 5 - \left(\frac{1}{3}\right)^2(9) $$ $$ = 5 - 1 = 4 $$ $$ l_{32} = \frac{1}{4} \left( 4 - \frac{2}{3}\cdot\frac{1}{3}\cdot9 \right) = \frac{1}{2} $$

Step 3

$$ d_3 = 10 - \left(\frac{2}{3}\right)^2(9) - \left(\frac{1}{2}\right)^2(4) $$ $$ = 10 - 4 - 1 = 5 $$

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Indefinite Matrix Example

$$ A = \begin{bmatrix} 1 & 2 & 0 \\ 2 & 1 & 2 \\ 0 & 2 & 1 \end{bmatrix} $$

Step 1

$$ d_1 = 1 $$ $$ l_{21} = 2, \quad l_{31}=0 $$

Step 2

$$ d_2 = 1 - (2)^2(1) $$ $$ = 1 - 4 = -3 $$ $$ l_{32} = \frac{1}{-3}(2) = -\frac{2}{3} $$

Step 3

$$ d_3 = 1 - \left(-\frac{2}{3}\right)^2(-3) $$ $$ = 1 + \frac{4}{3} = \frac{7}{3} $$

Since $d_2 < 0$, the matrix is indefinite.

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Comparison: LDL^T vs Cholesky

LDLT	Cholesky
$A = L D L^T$	$A = L L^T$
No square roots	Uses square roots
Works for indefinite matrices	Only positive definite

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Conclusion

• If all $d_i > 0$ → Positive definite matrix
• If some $d_i < 0$ → Indefinite matrix
• LDL^T avoids square roots

Questions on Triple Integration

20 Important Questions on Triple Integration

Basic Cartesian Integrals

1. Evaluate: $$\iiint_V (x+y+z)\, dV$$ where $0 \le x \le 1$, $0 \le y \le 2$, $0 \le z \le 3$.

2. Evaluate: $$\iiint_V xyz \, dV$$ where $0 \le x \le 2$, $0 \le y \le 1$, $0 \le z \le 3$.

3. Evaluate: $$\int_0^1 \int_0^{2x} \int_0^{x+y} dz\, dy\, dx$$

4. Change the order of integration: $$\int_0^1 \int_0^{1-x} \int_0^{1-x-y} f(x,y,z)\, dz\, dy\, dx$$

Region Between Surfaces

5. Evaluate: $$\iiint_V z \, dV$$ where $z = 0$ and $z = 4 - x^2 - y^2$.

6. Find the volume bounded by $$z = x^2 + y^2 \quad \text{and} \quad z = 4$$

7. Evaluate: $$\iiint_V (x^2+y^2)\, dV$$ inside cylinder $x^2 + y^2 = 9$, $0 \le z \le 5$.

8. Find the mass of the cube $0 \le x,y,z \le 1$ with density $\rho = x+y+z$.

Spherical Coordinates

9. Find the volume of sphere $$x^2+y^2+z^2 \le a^2$$

10. Evaluate: $$\iiint_V (x^2+y^2+z^2)\, dV$$ over sphere of radius $R$.

11. Find the volume of upper hemisphere $$x^2+y^2+z^2=16$$

Cylindrical Coordinates

12. Evaluate: $$\iiint_V r \, dV$$ where $x^2+y^2=4$, $0 \le z \le 3$.

13. Find volume enclosed by $$z=9-x^2-y^2$$ and $z=0$.

14. Evaluate: $$\iiint_V z\, dV$$ inside cylinder $x^2+y^2=1$, $0 \le z \le 2$.

Change of Variables

15. Evaluate using $u=x+y$, $v=x-y$: $$\iiint_V (x-y)\, dV$$

16. Evaluate: $$\iiint_V e^{-(x^2+y^2+z^2)} dV$$ over entire space.

Applications

17. Find centroid of solid $0 \le x \le a$, $0 \le y \le b$, $0 \le z \le c$.

18. Find moment of inertia about $z$-axis for cylinder $x^2+y^2 \le R^2$, $0 \le z \le h$.

19. Evaluate over tetrahedron $x=0$, $y=0$, $z=0$, $x+y+z=1$: $$\iiint_V xyz\, dV$$

20. Find volume common to sphere $$x^2+y^2+z^2=9$$ and cylinder $x^2+y^2=4$.

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Triple Integration

1. Introduction

Let $f(x,y,z)$ be a continuous function defined on a closed and bounded region $V \subset \mathbb{R}^3$. The triple integral of $f$ over $V$ is defined as

$$ \iiint_V f(x,y,z)\, dV $$

It represents the limit of Riemann sums:

$$ \iiint_V f(x,y,z)\, dV = \lim_{\max \Delta V_i \to 0} \sum f(x_i,y_i,z_i)\,\Delta V_i $$

provided the limit exists.

If $f(x,y,z)=1$, then the triple integral reduces to:

$$ \iiint_V 1\, dV = \text{Volume of } V $$

In practical computation, triple integrals are evaluated as iterated integrals:

$$ \iiint_V f(x,y,z)\, dV = \int_a^b \int_{g_1(x)}^{g_2(x)} \int_{h_1(x,y)}^{h_2(x,y)} f(x,y,z)\, dz\, dy\, dx $$

The order of integration may be changed whenever convenient.

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2. Explanation

Think of a double integral as adding up tiny rectangles to find the area of a region. A triple integral does the same thing in three dimensions — it adds up tiny boxes (small volumes) to find:

• Volume of a solid
• Mass (if density is given)
• Center of mass
• Physical quantities in engineering

Imagine breaking a solid object into many very small cubes. If we add up the value of the function at each cube times its small volume, we get the triple integral.

If we integrate just $1$, we simply get the volume:

$$ \iiint_V 1\, dV = \text{Volume} $$

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3. Geometric Interpretation (3D Illustration)

Small Volume Elements Inside a Solid:

Projection of a Solid onto the xy-plane:

Common Coordinate Systems:

• Cartesian: $dV = dx\,dy\,dz$
• Cylindrical: $dV = r\,dr\,d\theta\,dz$
• Spherical: $dV = \rho^2 \sin\phi\, d\rho\, d\phi\, d\theta$

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Summary

A triple integral allows us to measure quantities distributed throughout a three-dimensional region. It generalizes area (double integrals) to volume and physical applications in space.

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Triple Integrals – Solved Examples

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Example 1

Evaluate:

$$ \int_{0}^{1}\int_{0}^{1-y}\int_{0}^{1-y-z} z \, dx \, dz \, dy $$

Solution:

$$ = \int_{0}^{1}\int_{0}^{1-y} z(1-y-z)\, dz\, dy $$ $$ = \int_{0}^{1}\int_{0}^{1-y} \left[(1-y)z - z^2\right] dz\, dy $$ $$ = \int_{0}^{1} \left[ \frac{(1-y)z^2}{2} - \frac{z^3}{3} \right]_0^{1-y} dy $$ $$ = \int_{0}^{1} \left[ \frac{(1-y)^3}{2} - \frac{(1-y)^3}{3} \right] dy $$ $$ = \frac{1}{6}\int_{0}^{1}(1-y)^3 dy $$ $$ = \frac{1}{6}\left[\frac{(1-y)^4}{-4}\right]_0^1 $$ $$ = \frac{1}{24} $$

Geometric Region:

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Example 2

Evaluate:

$$ \iiint_V dV $$ where $$1 \le x \le 2,\quad 2 \le y \le 4,\quad 2 \le z \le 5.$$

Solution:

$$ = \int_{2}^{4}\int_{1}^{2}\int_{2}^{5} dz\, dx\, dy $$ $$ = \int_{2}^{4}\int_{1}^{2} 3\, dx\, dy $$ $$ = \int_{2}^{4} 3(1)\, dy $$ $$ = 3[y]_2^4 $$ $$ = 6 $$

Region (Rectangular Box):

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Example 3

Evaluate:

$$ \iiint_V 2x\, dV $$ where $V$ lies under the plane $$2x + 3y + z = 6$$ in the first octant.

Region Description:

$$ 0 \le z \le 6 - 2x - 3y $$ Projection on $xy$-plane: $$ 0 \le x \le 3, \quad 0 \le y \le \frac{6-2x}{3} $$

Integral:

$$ \int_0^3 \int_0^{(6-2x)/3} 2x(6-2x-3y)\, dy\, dx $$ $$ = \int_0^3 \left(\frac{4}{3}x^3 - 8x^2 + 12x\right) dx $$ $$ = 9 $$

Solid Figure:

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Example 4

Evaluate:

$$ \iiint_V z(x^2+y^2)\, dV $$ over the cylinder $$x^2+y^2 \le 1$$ between $z=2$ and $z=3$.

Integral Setup:

$$ = \int_{-1}^{1}\int_{-\sqrt{1-y^2}}^{\sqrt{1-y^2}} \int_2^3 z(x^2+y^2)\, dz\, dx\, dy $$ $$ = \frac{5}{2}\int_{-1}^{1}\int_{-\sqrt{1-y^2}}^{\sqrt{1-y^2}} (x^2+y^2)\, dx\, dy $$ Using $y=\sin\theta$, $$ = \frac{10}{3}\int_0^{\pi/2} \cos^4\theta\, d\theta + 10\int_0^{\pi/2} \sin^2\theta\cos^2\theta\, d\theta $$ $$ = \frac{5\pi}{4} $$

Cylinder Region:

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Final Answers

• Example 2: 1/24
• Example 3: 6
• Example 4: 9
• Example 5: 5π/4