Maths · Determining areas of the regions bounded by simple curves in standard forms
The area of the region bounded by the curve and between and is:
The area of the region bounded by the curve \( \boldsymbol{y}=\boldsymbol{x}^{2}+\mathbf{1} \) and \( \boldsymbol{y}=\mathbf{2} \boldsymbol{x}-\mathbf{2} \) between \( x=-1 \) and \( x=2 \) is:
- A. 9 sq . units
- B. 12sq. units
- C. 15 sq. units
- D. 14sq. units
Step-by-step solution
The area is found by integrating the difference of the functions from x=-1 to x=2. Since y=x^2+1 is above y=2x-2 in this interval, area = ∫_{-1}^{2} [(x^2+1)-(2x-2)] dx = ∫_{-1}^{2} (x^2-2x+3) dx = [x^3/3 - x^2 + 3x]_{-1}^{2} = (8/3-4+6) - (-1/3-1-3) = (14/3) - (-13/3) = 27/3 = 9.
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