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Business Mathematics and Statistics · Class 12 Commerce

Ch 3Integral Calculus – II (Area under curves; Application of Integration in Economics and Commerce) — Class 12 Business Mathematics and Statistics, concept-first.

!Figure 1 — Area Under y = x² from x = 1 to x = 3

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Area under a Curve

The area under a curve , bounded by the X-axis and two vertical lines and , is found by slicing the region into a very large number of extremely thin vertical strips, each of width and height , and adding up their areas.…

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Chapter contents

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1

Area Under a Curve Using Definite Integrals

!Figure 1 — Area Under y = x² from x = 1 to x = 3

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Cost and Revenue Functions from Marginal Functions

The marginal cost is the extra cost of producing one more unit at output level . Since integration reverses differentiation, the total cost function is recovered by integrating:

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Consumer's Surplus and Producer's Surplus

!Figure 2 — Demand and Supply Curves Meeting at Equilibrium (5, 20)

Exercises

+Show 4 questions4 questions
  1. Q7Find the area bounded by the curve $y=4x-x^2$ and the x-axis (i.e. between the curve's two x-intercepts).Free
  2. Q8The marginal cost of producing $x$ units is $MC=3x^2-4x+10$, and the fixed cost is ₹500. Find the total cost function and the cost of produc…Free
  3. Q9The marginal revenue is $MR=100-6x$. Find the total revenue function and the revenue from selling 8 units.Preview
  4. Q10For the demand function $p=40-x$ and supply function $p=10+2x$, find the equilibrium point, and then compute the Consumer's Surplus and Prod…Preview

Sample & Board Papers

Sample papers and previous-year board questions for this subject.

+Show 32 questions32 questions
  1. Q1Area bounded by $y = |x|$ between the limits $0$ and $2$ is : (a) $4$ sq.units (b) $1$ sq.unit (c) $3$ sq.units (d) $2$ sq.unitsPreview
  2. Q2The marginal cost function is $MC = 100\sqrt{x}$. Find AC, given that $TC = 0$ when the output is zero : (a) $\dfrac{200}{3x^{1/2}}$ (b) $\d…Preview
  3. Q3Find the area of the region bounded by the parabola $y = 4 - x^2$, $x$-axis and the lines $x = 0$, $x = 2$.Preview
  4. Q4The rate of new product is given by $f(x) = (100 + 2x^2)e^x$, where $x$ is the number of days the product is on the market. Find the total s…Preview
  5. Q5(a) The marginal cost $C'(x)$ and marginal revenue $R'(x)$ are given by $C'(x) = 50 + \dfrac{x}{50}$ and $R'(x) = 60$. The fixed cost is $\t…Preview
  6. Q6The profit of a function $p(x)$ is maximum when : (a) $MR = 0$ (b) $MC - MR = 0$ (c) $MC + MR = 0$ (d) $MC = 0$Preview
  7. Q7For a demand function p, if $\int \frac{dp}{p} = k \int \frac{dx}{x}$, then k is equal to : (a) $-\frac{1}{\eta_d}$ (b) $\eta_d$ (c) $\frac{…Preview
  8. Q8The producer's surplus when the supply function for a commodity is $p = 3 + x$ and $x_0 = 3$ is ______. (a) $\frac{3}{2}$ (b) $\frac{5}{2}$…Preview
  9. Q9Sketch the graph $y = |x + 3|$ and evaluate $\int_{-6}^{0} |x + 3|\, dx$.Preview
  10. Q10(a) The elasticity of demand with respect to price p for a commodity is $\eta_d = \frac{p + 2p^2}{100 - p - p^2}$. Find demand function wher…Preview
  11. Q11(a) Find the consumer's surplus and producer's surplus for the demand function $p_d = 25 - 3x$ and supply function $p_s = 5 + 2x$. OR (b) Co…Preview
  12. Q12The profit of a function $p(x)$ is maximum when : (a) $MR=0$ (b) $MC-MR=0$ (c) $MC+MR=0$ (d) $MC=0$Preview
  13. Q13The demand function for the marginal function $MR=100-9x^2$ is : (a) $100x-9x^2$ (b) $100-3x^2$ (c) $100+9x^2$ (d) $100x-3x^2$Preview
  14. Q14Find the area bounded by the curve $y=4x+3$ with $x$-axis between the lines $x=1$ and $x=4$.Preview
  15. Q15If $MR=20-5x+3x^2$, find total revenue function.Preview
  16. Q16(a) Find the consumer's surplus and producer's surplus for the demand function $P_d=25-3x$ and supply function $P_s=5+2x$. OR (b) X is norma…Preview
  17. Q17Area bounded by $y=|x|$ between the limits 0 and 2 is : (a) 2 sq. units (b) 1 sq. unit (c) 4 sq. units (d) 3 sq. unitsPreview
  18. Q18Area bounded by $y=e^x$ between the limits 0 to 1 is : (a) $\left(1-\frac{1}{e}\right)$ sq. units (b) $(e+1)$ sq. units (c) $(e-1)$ sq. unit…Preview
  19. Q19Calculate consumer's surplus if the demand function $p=122-5x-2x^2$ and $x=20$.Preview
  20. Q20Using integration find the area of the region bounded by the line $y-1=x$, the $x$-axis and the ordinates $x=-3$ and $x=3$.Preview
  21. Q21(a) The marginal cost $C'(x)$ and marginal revenue $R'(x)$ are given by $C'(x)=50+\frac{x}{50}$ and $R'(x)=60$. The fixed cost is ₹ 200. Det…Preview
  22. Q22If the marginal revenue of a firm is a constant, then the demand function is : (a) $C(x)$ (b) $MR$ (c) $AC$ (d) $MC$Preview
  23. Q23Area bounded by the curve $y=|x|$ between the limits $0$ and $2$ is : (a) $2$ sq. units (b) $1$ sq. unit (c) $4$ sq. units (d) $3$ sq. unitsPreview
  24. Q24If the marginal revenue function for a commodity is $MR=9-4x^{2}$, find the demand function.Preview
  25. Q25Calculate the area bounded by the parabola $y^{2}=4ax$ and its latus rectum.Preview
  26. Q26Using integration find the area of the circle whose centre is at the origin and the radius is 5 units.Preview
  27. Q27(a) A firm has the marginal revenue function given by $MR=\dfrac{a}{(x+b)^{2}}-c$. Where $x$ is the output and $a,b,c$ are constants. Show t…Preview
  28. Q28The profit of a function $p(x)$ is maximum when : (a) $MR = 0$ (b) $MC - MR = 0$ (c) $MC + MR = 0$ (d) $MC = 0$Preview
  29. Q29The area bounded by the parabola $y^2 = 4x$ bounded by its latus rectum is : (a) $\frac{72}{3}$ sq.units (b) $\frac{16}{3}$ sq.units (c) $\f…Preview
  30. Q30The Marginal Cost function $MC = 14 - 6x + 4x^2$. Then find the cost function if fixed cost is 12.Preview
  31. Q31Find the area bounded by the parabola $y = x^2$ and the line $y = 4$.Preview
  32. Q32(a) The demand and supply function of a commodity are $P_d = 18 - 2x - x^2$ and $P_s = 2x - 3$. Find the consumer's surplus and producer's s…Preview

More questions

+Show 6 questions6 questions
  1. Example 1Find the area bounded by the curve $y=x^2$, the x-axis, and the lines $x=1$ and $x=3$.Free
  2. Example 2Find the area under the line $y=3x+2$ between $x=0$ and $x=4$.Free
  3. Example 3The marginal cost of producing $x$ units is $MC=6x+5$ (in ₹ per unit), and the fixed cost is ₹200. Find the total cost function, and the cos…Preview
  4. Example 4The marginal revenue for a firm's product is $MR=50-4x$. Find the total revenue function, and the revenue from selling 5 units.Preview
  5. Example 5For the demand function $p=30-2x$ and supply function $p=5+3x$, find the equilibrium point, and then compute the Consumer's Surplus and Prod…Preview
  6. Example 6Find the area between the curves $y=x$ and $y=x^2$ from $x=0$ to $x=1$.Preview