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Area between two curves worksheets Grade 12

Find where two curves cross, then integrate top minus bottom.

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How to do it

Find where the curves cross. Integrate (top − bottom) between those x values.

the area between y=−x2+5x−1 and y=2x+1:16\text{the area between }y = -x^{2} + 5x - 1\ \text{and }y = 2x + 1:\quad \dfrac{1}{6}
−x2+5x−1=2x+1 when x2−3x+2=0:x=1 and x=2-x^{2} + 5x - 1 = 2x + 1\ \text{when }x^{2} - 3x + 2 = 0: x = 1\ \text{and }x = 2where they cross
area =∫12(−x2+3x−2) dx\text{area }= \int_{1}^{2} (-x^{2} + 3x - 2)\, dxtop minus bottom
F(x)=−x33+3x22−2xF(x) = -\dfrac{x^{3}}{3} + \dfrac{3x^{2}}{2} - 2x
(−23)−(−56)=16(-\dfrac{2}{3}) - (-\dfrac{5}{6}) = \dfrac{1}{6}F(2) − F(1)

Example problems

  1. the area between y=−2x2+7x−8 and y=−3x+4:\text{the area between }y = -2x^{2} + 7x - 8\ \text{and }y = -3x + 4:\quad
  2. the area between y=−2x2+13x−18 and y=x−2:\text{the area between }y = -2x^{2} + 13x - 18\ \text{and }y = x - 2:\quad
  3. the area between y=−3x2+6x+38 and y=3x+2:\text{the area between }y = -3x^{2} + 6x + 38\ \text{and }y = 3x + 2:\quad
  4. the area between y=−2x2+x+8 and y=3x+4:\text{the area between }y = -2x^{2} + x + 8\ \text{and }y = 3x + 4:\quad

Integrals

  1. Antiderivatives with the power rule
  2. Integrals of 1/x, eˣ, sin and cos
  3. Integration by u-substitution
  4. Evaluating definite integrals
  5. Area under a curve
  6. Area between two curves
  7. Average value of a function