Definite Integrals & Area: Calculating Areas Under Curves & Between Functions

TAKEAWAYS:

• Definite integrals compute exact areas under curves, unlike approximations from Riemann sums.
• The Fundamental Theorem of Calculus connects integrals and derivatives to evaluate areas efficiently.
• Areas between curves require subtracting integrals of the upper and lower functions.
• Symmetry (even/odd functions) can simplify calculations.
• Applications include physics (work, pressure), economics (profit), and biology (growth rates).
• NEB exams test both computation and conceptual understanding of area interpretation.

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1. Definite Integrals: The Area Under a Curve

What is a Definite Integral?

A definite integral calculates the signed area between a curve $y=f(x)$ and the x-axis from $x=a$ to $x=b$. It is written as: $${\int }_a^{b}f(x)dx$$

• If $f(x)$ is above the x-axis, the area is positive.
• If $f(x)$ is below the x-axis, the area is negative.
• The total area (always positive) requires taking the absolute value or splitting the integral where the curve crosses the x-axis.

How is it Calculated?

The Fundamental Theorem of Calculus (FTC) connects integrals and derivatives: $${\int }_a^{b}f(x)dx=F(b)-F(a)$$ where $F(x)$ is the antiderivative of $f(x)$.

Worked Example 1: Basic Definite Integral

Problem: Find the area under $f(x)=3x^2$ from $x=1$ to $x=3$.

Solution:

1. Find the antiderivative $F(x)$: $$F(x)=\int 3x^{2}dx=x^3+C$$
2. Apply the FTC: $${\int }_1^{3}3x^{2}dx=F(3)-F(1)=(3{)}^3-(1{)}^3=27-1=26$$
3. Interpretation: The area under $f(x)=3x^2$ from $x=1$ to $x=3$ is 26 square units.

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2. Riemann Sums vs. Definite Integrals

Riemann Sums: Approximating Areas

Before calculus, mathematicians used Riemann sums to approximate areas by dividing the region into rectangles.

• Left Riemann Sum: Uses the left endpoint of each subinterval.
• Right Riemann Sum: Uses the right endpoint.
• Midpoint Riemann Sum: Uses the midpoint of each subinterval.

As the number of rectangles ($n$) increases, the approximation becomes more accurate.

Comparison Table

Worked Example 2: Riemann Sum Approximation

Problem: Approximate the area under $f(x)=x^2$ from $x=0$ to $x=2$ using 4 equal subintervals (Left Riemann Sum).

Solution:

1. Divide $[0,2]$ into 4 subintervals: $\Delta x=\frac{2-0}{4}=0.5$.
2. Evaluate $f(x)$ at left endpoints: $x_0=0,x_1=0.5,x_2=1,x_3=1.5$.
3. Compute the sum: $$\text{Area}\approx (0.5{)}^2(0.5)+(1{)}^2(0.5)+(1.5{)}^2(0.5)+(2{)}^2(0.5)$$ $$=0.125+0.5+1.125+2=3.75$$
4. Exact area (using integral): $${\int }_0^2{x}^{2}dx={\frac{x^3}{3}|}_0^2=\frac{8}{3}\approx \mathrm{2.666...}$$ Observation: The Left Riemann Sum overestimates because $f(x)$ is increasing.

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3. Areas Between Two Curves

To find the area between two curves $y=f(x)$ (upper) and $y=g(x)$ (lower) from $x=a$ to $x=b$, use: $$\text{Area}={\int }_a^b[f(x)-g(x)]dx$$

Key Steps:

1. Sketch the graphs to identify which function is upper and which is lower.
2. Find points of intersection (if any) to determine limits.
3. Set up the integral as $\int (\text{Upper}-\text{Lower})dx$.

Worked Example 3: Area Between Curves

Problem: Find the area between $f(x)=x^2+2$ and $g(x)=x+4$ from $x=0$ to $x=2$.

Solution:

1. Sketch the graphs (see below).
2. Check which is upper/lower:
   • At $x=1$, $f(1)=3$, $g(1)=5$. So, $g(x)$ is upper.
3. Set up the integral: $$\text{Area}={\int }_0^2[g(x)-f(x)]dx={\int }_0^2[(x+4)-(x^2+2)]dx$$ $$={\int }_0^2(-x^2+x+2)dx$$
4. Compute the integral: $$={\left[-\frac{x^3}{3}+\frac{x^2}{2}+2x\right]}_0^2$$ $$=\left(-\frac{8}{3}+2+4\right)-(0)=\frac{10}{3}\approx 3.33$$

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4. Special Cases: When Curves Cross

If the curves intersect, split the integral at the points of intersection.

Worked Example 4: Curves Crossing

Problem: Find the area between $f(x)=x^2$ and $g(x)=2x$.

Solution:

1. Find intersection points: $$x^2=2x⟹x(x-2)=0⟹x=0,2$$
2. Determine upper/lower:
   • For $0<x<2$, $2x>x^2$. So, $g(x)$ is upper.
3. Set up the integral: $$\text{Area}={\int }_0^2(2x-x^2)dx$$
4. Compute: $$={\left[x^2-\frac{x^3}{3}\right]}_0^2=(4-\frac{8}{3})-(0)=\frac{4}{3}$$

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5. Symmetry in Definite Integrals

Even and Odd Functions

• Even function: $f(-x)=f(x)$ (symmetric about y-axis). $${\int }_{-a}^{a}f(x)dx=2{\int }_0^{a}f(x)dx$$
• Odd function: $f(-x)=-f(x)$ (symmetric about origin). $${\int }_{-a}^{a}f(x)dx=0$$

Worked Example 5: Using Symmetry

Problem: Evaluate ${\int }_{-2}^2(x^3+\cos x)dx$.

Solution:

1. Split the integral: $${\int }_{-2}^2{x}^{3}dx+{\int }_{-2}^2\cos xdx$$
2. Analyze each part:
   • $x^3$ is odd → integral = 0.
   • $\cos x$ is even → $2{\int }_0^2\cos xdx$.
3. Compute: $$2{\left[\sin x\right]}_0^2=2(\sin 2-\sin 0)=2\sin 2$$

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6. Applications of Definite Integrals

Definite integrals are used in:

1. Physics:
   • Work done by a variable force: $W=\int F(x)dx$.
   • Pressure in fluids: $P=\int \rho ghdA$.
2. Economics:
   • Consumer surplus = $\int (\text{Demand}-\text{Price})dx$.
3. Biology:
   • Population growth under varying rates.

Worked Example 6: Physics Application

Problem: A force $F(x)=5x+2$ acts on an object from $x=1$ to $x=3$. Find the work done.

Solution: $$W={\int }_1^3(5x+2)dx={\left[\frac{5x^2}{2}+2x\right]}_1^3$$ $$=\left(\frac{45}{2}+6\right)-\left(\frac{5}{2}+2\right)=\frac{45}{2}+4-\frac{9}{2}=22\text{ Joules}$$

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7. Common Mistakes to Avoid

1. Forgetting absolute value when area is entirely below the x-axis.
2. Mixing upper/lower functions in area-between-curves problems.
3. Incorrect limits when curves intersect.
4. Ignoring units (area should be in square units).
5. Misapplying symmetry (check if function is even/odd first).

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Exam Tip: How NEB Tests This Unit

NEB exams on Definite Integrals and Area typically include:

1. Direct computation of definite integrals (3–5 marks).
2. Area under a single curve (3–5 marks).
3. Area between two curves (5–7 marks).
4. Application-based problems (physics/economics, 5–8 marks).
5. Riemann sum approximations (3–5 marks).

NEB-Style Questions

Question 1 (Direct Integral)

Evaluate: $${\int }_1^4(2x^2-3x+1)dx$$

Question 2 (Area Under Curve)

Find the area bounded by $y=\sqrt{x}$, the x-axis, and the lines $x=1$ and $x=4$.

Question 3 (Area Between Curves)

Find the area enclosed by $y=x^2$ and $y=4$.

Question 4 (Application)

A tank has a cross-sectional area $A(x)=x^2+1$ at height $x$. Find the volume if the tank is 5 units tall.

Question 5 (Riemann Sum)

Approximate ${\int }_0^3{x}^{2}dx$ using 3 equal subintervals (Right Riemann Sum).

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Final Advice:

• Always sketch graphs before setting up integrals.
• Double-check upper/lower functions in area problems.
• Practice symmetry shortcuts to save time.
• Memorize the Fundamental Theorem of Calculus—it’s the key to solving integrals!

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Graphs of f(x) = x² (even) and f(x) = x³ (odd) (Image: Saracena9000, CC0, via Wikimedia Commons)