Applications of Integration: Areas & Volumes via Slicing

Core Concepts

1. Area Under a Curve (Single Integral)

Definition

The area $A$ between a curve $y=f(x)$, the x-axis, and vertical lines $x=a$ and $x=b$ is given by: $$A={\int }_a^b|f(x)|dx$$

• Key Notes:
  • If $f(x)\ge 0$ on $[a,b]$, the absolute value can be omitted.
  • For curves below the x-axis, integrate $-f(x)$ or use $|f(x)|$.

How It Works

1. Partition the Interval: Divide $[a,b]$ into $n$ subintervals of width $\Delta x$.
2. Approximate Area: Use rectangles (left/right/midpoint Riemann sums) to estimate area.
3. Take the Limit: As $n\to \infty$, the sum becomes the definite integral.

Worked Example: Area Between Curves

Problem: Find the area bounded by $y=x^2$ and $y=2x-x^2$ from $x=0$ to $x=2$.

Solution:

1. Find Points of Intersection: Solve $x^2=2x-x^2$: $$2x^2-2x=0⟹x(2x-2)=0⟹x=0\text{ or }x=1.$$ The curves intersect at $x=0$ and $x=1$.

2. Set Up the Integral: The upper curve is $y=2x-x^2$ and the lower curve is $y=x^2$. $$A={\int }_0^1\left[(2x-x^2)-(x^2)\right]dx={\int }_0^1(2x-2x^2)dx.$$

3. Compute the Integral: $$A={\left[x^2-\frac{2x^3}{3}\right]}_0^1=\left(1-\frac{2}{3}\right)-0=\frac{1}{3}\text{ square units}.$$

Comparison Table: Area Methods

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2. Volume of Solids of Revolution

Methods

Volumes are calculated by rotating a region about the x-axis or y-axis. Three primary methods exist:

A. Disk Method

• Use Case: Rotating around the x-axis or y-axis where the cross-section is a disk.
• Formula:
  • Rotation about x-axis: $$V=\pi {\int }_a^b[f(x){]}^{2}dx.$$
  • Rotation about y-axis (if $x=g(y)$): $$V=\pi {\int }_c^d[g(y){]}^{2}dy.$$

B. Washer Method

• Use Case: When there is a "hole" in the solid (e.g., rotating between two curves).
• Formula: $$V=\pi {\int }_a^b\left([f_{\text{outer}}(x){]}^2-[f_{\text{inner}}(x){]}^2\right)dx.$$

C. Shell Method

• Use Case: When rotating around a vertical or horizontal line not on the curve (e.g., $x=-1$).
• Formula:
  • Rotation about y-axis: $$V=2\pi {\int }_a^{b}x\cdot f(x)dx.$$
  • Rotation about x-axis: $$V=2\pi {\int }_c^{d}y\cdot g(y)dy.$$

Worked Example: Disk Method

Problem: Find the volume obtained by rotating $y=\sqrt{x}$ about the x-axis from $x=0$ to $x=4$.

Solution:

1. Identify the Radius: $r(x)=\sqrt{x}$.
2. Set Up the Integral: $$V=\pi {\int }_0^4(\sqrt{x}{)}^{2}dx=\pi {\int }_0^{4}xdx.$$
3. Compute the Integral: $$V=\pi {\left[\frac{x^2}{2}\right]}_0^4=\pi \left(\frac{16}{2}-0\right)=8\pi \text{ cubic units}.$$

Worked Example: Washer Method

Problem: Find the volume obtained by rotating the region bounded by $y=x^2$ and $y=4$ about the x-axis.

Solution:

1. Find Points of Intersection: Solve $x^2=4⟹x=\pm 2$. Use $x=0$ to $x=2$ (symmetry).
2. Outer Radius: $R(x)=4$. Inner Radius: $r(x)=x^2$.
3. Set Up the Integral: $$V=\pi {\int }_0^2\left(4^2-(x^2{)}^2\right)dx=\pi {\int }_0^2(16-x^4)dx.$$
4. Compute the Integral: $$V=\pi {\left[16x-\frac{x^5}{5}\right]}_0^2=\pi \left(32-\frac{32}{5}\right)=\frac{128\pi }{5}\text{ cubic units}.$$

Worked Example: Shell Method

Problem: Find the volume obtained by rotating $y=x^2$ about the y-axis from $x=0$ to $x=2$.

Solution:

1. Identify Shell Dimensions:
   • Radius: $x$.
   • Height: $y=x^2$.
2. Set Up the Integral: $$V=2\pi {\int }_0^{2}x\cdot x^{2}dx=2\pi {\int }_0^2{x}^{3}dx.$$
3. Compute the Integral: $$V=2\pi {\left[\frac{x^4}{4}\right]}_0^2=2\pi \left(\frac{16}{4}-0\right)=8\pi \text{ cubic units}.$$

Comparison Table: Volume Methods

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3. Numerical Approximation of Areas

When exact integrals are difficult, numerical methods approximate areas using rectangles or trapezoids.

A. Rectangle Method (Riemann Sums)

• Left Endpoint Approximation: $$A\approx \Delta x\sum _{i=0}^{n-1}f(x_i),\Delta x=\frac{b-a}{n}.$$
• Right Endpoint Approximation: $$A\approx \Delta x\sum _{i=1}^{n}f(x_i).$$

Worked Example: Rectangle Method

Problem: Estimate the area under $y=x^2$ from $x=0$ to $x=1$ using 4 subintervals (left endpoints).

Solution:

1. Partition: $\Delta x=\frac{1-0}{4}=0.25$. Points: $x_0=0,x_1=0.25,x_2=0.5,x_3=0.75$.
2. Compute Heights: $f(x_0)=0$, $f(x_1)=(0.25{)}^2=0.0625$, $f(x_2)=(0.5{)}^2=0.25$, $f(x_3)=(0.75{)}^2=0.5625$.
3. Sum Areas: $$A\approx 0.25\times (0+0.0625+0.25+0.5625)=0.25\times 0.875=\mathrm{0.21875.}$$ (Exact area: ${\int }_0^1{x}^{2}dx=\frac{1}{3}\approx 0.333$.)

B. Trapezoidal Rule

• Formula: $$A\approx \frac{\Delta x}{2}\left[f(x_0)+2f(x_1)+2f(x_2)+\cdots +2f(x_{n-1})+f(x_n)\right].$$

Worked Example: Trapezoidal Rule

Problem: Estimate the area under $y=x^2$ from $x=0$ to $x=1$ using 4 subintervals.

Solution:

1. Partition: Same as above.
2. Apply Trapezoidal Rule: $$A\approx \frac{0.25}{2}\left[0+2(0.0625+0.25+0.5625)+1\right]=\frac{0.25}{2}\left[0+2(0.875)+1\right]=\frac{0.25}{2}\times 2.75=\mathrm{0.34375.}$$ (Closer to the exact value of $\frac{1}{3}$.)

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4. Practical Applications

A. Optimization Problems

Example: Maximizing Area with Fixed Perimeter (Fencing Problem) Problem: A farmer has 2000 ft of fencing to enclose a rectangular field adjacent to a river (no fence needed on one side). Find the dimensions for maximum area.

Solution:

1. Define Variables: Let $x$ = length parallel to the river, $y$ = length perpendicular to the river. Perimeter constraint: $x+2y=2000⟹y=\frac{2000-x}{2}$.
2. Area Function: $$A(x)=x\cdot y=x\left(\frac{2000-x}{2}\right)=1000x-\frac{x^2}{2}.$$
3. Find Critical Points: $$A^{\prime }(x)=1000-x=0⟹x=1000.$$ Then $y=\frac{2000-1000}{2}=500$.
4. Conclusion: The maximum area is achieved with dimensions 1000 ft × 500 ft, yielding an area of 500,000 sq ft.

B. Area Between Curves

Example: Area Between $y^2=x$ and $x=2$ Problem: Estimate the area bounded by $y^2=x$, $x=0$, and $x=2$.

Solution:

1. Express as Function of $y$: $x=y^2$. The curves intersect at $y=\pm \sqrt{2}$.
2. Set Up Integral: $$A={\int }_{-\sqrt{2}}^{\sqrt{2}}(2-y^2)dy.$$
3. Compute: $$A=2{\int }_0^{\sqrt{2}}(2-y^2)dy=2{\left[2y-\frac{y^3}{3}\right]}_0^{\sqrt{2}}=2\left(2\sqrt{2}-\frac{2\sqrt{2}}{3}\right)=\frac{8\sqrt{2}}{3}.$$

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Exam Tip

Common Pitfalls & Strategies

1. Identify Bounds Correctly:

   • Always find points of intersection for area between curves.
   • For volumes, ensure correct limits (e.g., $x=a$ to $x=b$ vs. $y=c$ to $y=d$).

2. Disk vs. Washer vs. Shell:

   • Disk: Use when rotating a single function about an axis it crosses.
   • Washer: Use when there’s a hole (two functions).
   • Shell: Use when rotating about a vertical/horizontal line not on the curve (e.g., $x=-1$).

3. Numerical Methods:

   • Left/right endpoint approximations are less accurate than the trapezoidal rule.
   • For exams, show all steps (partition, heights, sums).

4. Units:

   • Areas are in square units (e.g., ft², m²).
   • Volumes are in cubic units (e.g., ft³, m³).

5. Optimization:

   • Always verify critical points (e.g., second derivative or endpoints).
   • Constraints (like perimeter) must be incorporated into the objective function.

6. Graphical Intuition:

   • Sketch the region/curve before setting up integrals. Label axes and bounds clearly.

High-Score Techniques

• Label Diagrams: Draw the region and shade the area/volume being calculated.
• Show Work: Even for numerical methods, write out the partition and sum explicitly.
• Check Units: Ensure your final answer has the correct units (e.g., "square meters").
• Practice Past Questions: Focus on mixing area and volume problems, as they often appear together.

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