Skip to content
Worked Examples · Example 17

Q.Find the consumers' surplus for the demand function p=25−x−x2p = 25 - x - x^2 when p0=19p_0 = 19.

Yanam CbseNCERTSubjective· 3mImportance★★★★★
29% · 17/59 Questions
✓ Free question

Find the quantity x0x_0 at p0=19p_0=19, then CS=∫0x0p dx−p0x0=223≈₹7.33CS=\int_0^{x_0} p\,dx - p_0x_0=\dfrac{22}{3}\approx ₹7.33.

Consumer's surplus CS=∫0x0f(x) dx−p0x0CS=\displaystyle\int_0^{x_0} f(x)\,dx - p_0x_0, where p=f(x)p=f(x) is the demand curve, p0p_0 the market price and x0x_0 the quantity demanded at p0p_0.

  1. Demand: p=25−x−x2p=25-x-x^2. Set p=p0=19p=p_0=19: 19=25−x−x2⇒x2+x−6=0.19=25-x-x^2\Rightarrow x^2+x-6=0.
  2. Factor: (x+3)(x−2)=0⇒x=2(x+3)(x-2)=0\Rightarrow x=2 (reject x=−3x=-3). So x0=2x_0=2.
  3. ∫02(25−x−x2) dx=[25x−x22−x33]02=50−2−83=48−83=1363.\displaystyle\int_0^{2}(25-x-x^2)\,dx=\Big[25x-\tfrac{x^2}{2}-\tfrac{x^3}{3}\Big]_0^2=50-2-\tfrac{8}{3}=48-\tfrac{8}{3}=\tfrac{136}{3}.
  4. p0x0=19×2=38.p_0x_0=19\times2=38.
  5. CS=1363−38=136−1143=223≈7.33.CS=\dfrac{136}{3}-38=\dfrac{136-114}{3}=\dfrac{22}{3}\approx 7.33.
✓Final answer

Consumer's surplus =223≈₹7.33=\dfrac{22}{3}\approx ₹7.33.

Unlock everything free for 14 days

  • Full step-by-step solutions
  • Concept-first explanations
  • Methods, shortcuts & mistakes
  • PYQ mapping + timed mock tests

Full access for 14 days. No credit card required.