Integration & Business Applications: Area, Volume, Cost, Revenue, Profit

TAKEAWAYS:

• Integration finds the total quantity from a rate (e.g., total cost from marginal cost, total revenue from marginal revenue).
• Business applications include calculating profit, consumer surplus, and depreciation using integrals.
• Area under a curve = ∫f(x)dx between limits (e.g., total revenue from demand curve).
• Volume of revolution uses the disk/washer method for storage tanks or containers.
• Marginal functions (cost, revenue, profit) are integrated to find totals.
• Exam focus: Always check units (e.g., Rs/unit → Rs) and limits (upper/lower bounds).

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1. What is Integration? (Recap from Class 11)

Integration is the reverse of differentiation. If differentiation gives the rate of change (slope), integration gives the total accumulated quantity.

Example 1: Find ∫(3x² + 2x + 1)dx Solution:

∫(3x² + 2x + 1)dx = 3∫x²dx + 2∫xdx + ∫1dx
                 = 3(x³/3) + 2(x²/2) + x + C
                 = x³ + x² + x + C

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2. Area Under a Curve (Definite Integral)

The area under a curve y = f(x) between x = a and x = b is given by:

∫[a to b] f(x) dx

Key Points:

• If f(x) is above the x-axis, the area is positive.
• If f(x) is below the x-axis, the area is negative (take absolute value for total area).
• Business use: Total revenue, total cost, or total profit from marginal functions.

Example 2: Find the area under y = √x from x = 1 to x = 4. Solution:

∫[1 to 4] √x dx = ∫[1 to 4] x^(1/2) dx
               = [x^(3/2)/(3/2)] from 1 to 4
               = (2/3)(4^(3/2) - 1^(3/2))
               = (2/3)(8 - 1) = 14/3 ≈ 4.67 square units

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3. Business Applications of Integration

(A) Total Cost from Marginal Cost (MC)

• Marginal Cost (MC): Cost of producing one more unit.
• Total Cost (TC): Integral of MC.

TC = ∫[0 to Q] MC(x) dx + Fixed Cost (FC)

Example 3: If MC = 2x + 10 and FC = 50, find TC for Q = 5. Solution:

TC = ∫[0 to 5] (2x + 10) dx + 50
   = [x² + 10x] from 0 to 5 + 50
   = (25 + 50) - (0 + 0) + 50 = 125 Rs

(B) Total Revenue from Marginal Revenue (MR)

• Marginal Revenue (MR): Revenue from selling one more unit.
• Total Revenue (TR): Integral of MR.

TR = ∫[0 to Q] MR(x) dx

Example 4: If MR = 50 - 2x, find TR for Q = 10. Solution:

TR = ∫[0 to 10] (50 - 2x) dx
   = [50x - x²] from 0 to 10
   = (500 - 100) - (0 - 0) = 400 Rs

(C) Total Profit

Profit = TR - TC = ∫MR dx - ∫MC dx - FC

Example 5: If MR = 100 - x and MC = 20 + x, FC = 20, find profit at Q = 10. Solution:

TR = ∫(100 - x)dx = [100x - x²/2] from 0 to 10 = 1000 - 50 = 950
TC = ∫(20 + x)dx + 20 = [20x + x²/2] from 0 to 10 + 20 = 200 + 50 + 20 = 270
Profit = 950 - 270 = 680 Rs

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4. Volume of Revolution (Disk Method)

Used to find the volume of a solid formed by rotating a curve around the x-axis or y-axis. Formula:

V = π ∫[a to b] [f(x)]² dx  (Rotation around x-axis)

Example 6: Find the volume when y = √x is rotated from x = 1 to x = 4. Solution:

V = π ∫[1 to 4] (√x)² dx = π ∫[1 to 4] x dx
  = π [x²/2] from 1 to 4
  = π (8 - 0.5) = 7.5π ≈ 23.56 cubic units

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5. Consumer and Producer Surplus

(A) Consumer Surplus (CS)

• Area between demand curve and equilibrium price.

CS = ∫[Pe to Pmax] D(x) dx

Example 7: Demand curve D(x) = 50 - 2x, equilibrium price Pe = 20. Find CS. Solution:

Find Q at Pe=20: 20 = 50 - 2Q → Q = 15
CS = ∫[20 to 50] (50 - 2x)/2 dx (since D(x) = P)
   = (1/2) ∫(50 - 2x) dx from Q=0 to Q=15
   = (1/2) [50x - x²] from 0 to 15
   = (1/2) (750 - 225) = 262.5 Rs

(B) Producer Surplus (PS)

• Area between supply curve and equilibrium price.

PS = ∫[0 to Qe] S(x) dx

Example 8: Supply curve S(x) = 10 + x, Pe = 20. Find PS. Solution:

Find Q at Pe=20: 20 = 10 + Q → Q = 10
PS = ∫[10 to 20] (x - 10) dx
   = [x²/2 - 10x] from 10 to 20
   = (200 - 200) - (50 - 100) = 50 Rs

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6. Depreciation Using Integration

Depreciation is the loss in value of an asset over time. If the rate of depreciation is given, we can find the total depreciation using integration.

Example 9: A machine depreciates at a rate of D(t) = 1000 - 20t (in Rs/year). Find total depreciation from t=0 to t=5. Solution:

Total Depreciation = ∫[0 to 5] (1000 - 20t) dt
                   = [1000t - 10t²] from 0 to 5
                   = (5000 - 250) - (0 - 0) = 4750 Rs

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7. Comparison Table: Integration vs. Differentiation

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8. Common Mistakes to Avoid

1. Forgetting limits: Always check upper and lower bounds in definite integrals.
2. Ignoring fixed costs: TC = ∫MC dx + FC.
3. Sign errors: If the curve is below the x-axis, the integral gives a negative area (take absolute value if needed).
4. Unit mismatch: Ensure units are consistent (e.g., Rs/unit → Rs).
5. Incorrect antiderivative: Double-check integration rules (e.g., ∫xⁿ dx = xⁿ⁺¹/(n+1) + C).

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Exam Tip

1. Always label your answer: Write "Total Cost = ..." or "Area = ..." before giving the numerical value.
2. Show all steps: Even if a question seems simple, write the integral, antiderivative, and evaluation clearly.
3. Check units: If the answer is in Rs, ensure your integral was of a function in Rs/unit.
4. Practice graph interpretation: Sketch the curve and shade the area you are calculating.
5. Memorize key formulas:
   • Area = ∫f(x)dx
   • Volume (disk) = π ∫[f(x)]² dx
   • Profit = ∫(MR - MC)dx - FC
6. NEB-style questions often test:
   • Calculating total cost/revenue from marginal functions.
   • Finding areas under demand/supply curves (CS/PS).
   • Volume of revolution for business storage tanks.

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NEB Board-Style Questions (Practice)

Short Answer (5 marks each)

1. The marginal cost of producing x units is MC = 3x² + 2x + 10. If fixed cost is Rs. 50, find the total cost when 5 units are produced.
2. Find the area under the curve y = 3x² + 2 from x = 1 to x = 3.
3. The demand function is P = 100 - 2x. Find the consumer surplus when the equilibrium price is Rs. 60.
4. A company’s marginal revenue is MR = 50 - x. Find the total revenue when 10 units are sold.
5. Find the volume generated by rotating y = √x from x = 0 to x = 4 about the x-axis.

Long Answer (10 marks each)

6. The marginal cost of a product is MC = 2x + 5 and marginal revenue is MR = 50 - x. Fixed cost is Rs. 20.
   • Find the total cost and total revenue when 10 units are produced.
   • Calculate the profit at this level of production.
7. The demand function for a product is P = 20 - 0.5x and supply function is P = 5 + 0.25x.
   • Find the equilibrium price and quantity.
   • Calculate the consumer and producer surplus at equilibrium.
8. A storage tank is formed by rotating y = 2√x from x = 1 to x = 4 about the x-axis. Find its volume.

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Summary

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Final Note: Integration is like "adding up tiny slices" to find the whole. In business, it helps answer:

• How much total cost/revenue/profit?
• What is the area under a demand curve (surplus)?
• How much volume does a storage tank hold?

Practice integrating marginal functions and real-world curves to master this unit!