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Area under a curve worksheets Grade 12

Find the area between a curve and the x-axis with a definite integral.

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How to do it

When the curve is above the x-axis, the area is the definite integral from a to b.

the area under y=2x2+x+4 from x=1 to x=3:883\text{the area under }y = 2x^{2} + x + 4\ \text{from }x = 1\ \text{to }x = 3:\quad \dfrac{88}{3}
area =∫13(2x2+x+4) dx\text{area }= \int_{1}^{3} (2x^{2} + x + 4)\, dxthe curve is above the x-axis here
F(x)=2x33+x22+4xF(x) = \dfrac{2x^{3}}{3} + \dfrac{x^{2}}{2} + 4x
(692)−(316)=883(\dfrac{69}{2}) - (\dfrac{31}{6}) = \dfrac{88}{3}

Example problems

  1. the area under y=3x2−2x+9 from x=2 to x=4:\text{the area under }y = 3x^{2} - 2x + 9\ \text{from }x = 2\ \text{to }x = 4:\quad
  2. the area under y=3x2+4x+2 from x=1 to x=5:\text{the area under }y = 3x^{2} + 4x + 2\ \text{from }x = 1\ \text{to }x = 5:\quad
  3. the area under y=2x2−x+9 from x=0 to x=4:\text{the area under }y = 2x^{2} - x + 9\ \text{from }x = 0\ \text{to }x = 4:\quad
  4. the area under y=3x2+3 from x=1 to x=4:\text{the area under }y = 3x^{2} + 3\ \text{from }x = 1\ \text{to }x = 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