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NTA Abhyas 2020
Exam
NTA Abhyas
Year
2020
Subject
Mathematics
Chapter
Area Under Curves

NTA Abhyas 2020 · Mathematics · Area Under Curves

The area bounded by the graph y = | [x - 3]| , above the x -axis...

The area bounded by the graph y=∣[x−3]∣y = \left| \left[x - 3\right]\right|, above the xx-axis from x=−2x = - 2 to x=3x = 3 is ([.]denotesthegreatestintegerfunction)(\left[.\right] denotes the greatest integer function)

Options

  1. A7sq.units7sq. units
  2. B15sq.units15sq. units
  3. C21sq.units21sq. units
  4. D28sq.units28sq. units

Correct answer

B. 15sq.units15sq. units

Step-by-step solution

∴∴ Required area

=∫−23∣[x−3]∣dx= \mathop{∫}_{- 2}^{3}\left|\left[x - 3\right]\right|dx

=∫−2−1∣[x−3]∣dx+∫−10∣[x−3]∣dx= \mathop{∫}_{- 2}^{- 1}\left|\left[x - 3\right]\right|dx + \mathop{∫}_{- 1}^{0}\left|\left[x - 3\right]\right|dx

+∫01∣[x−3]∣dx+∫12∣[x−3]∣dx+∫23∣[x−3]∣dx+ \mathop{∫}_{0}^{1}\left|\left[x - 3\right]\right|dx + \mathop{∫}_{1}^{2}\left|\left[x - 3\right]\right|dx + \mathop{∫}_{2}^{3}\left|\left[x - 3\right]\right|dx

=∫−2−15dx+∫−104.dx+∫013.dx+∫122.dx+∫231.dx= \mathop{∫}_{- 2}^{- 1}5dx + \mathop{∫}_{- 1}^{0}4. dx + \mathop{∫}_{0}^{1}3. dx + \mathop{∫}_{1}^{2}2. dx + \mathop{∫}_{2}^{3}1. dx

=5(1)+4(1)+3(1)+2(1)+1(1)= 5 \left(1\right) + 4 \left(1\right) + 3 \left(1\right) + 2\left(1\right) + 1\left(1\right)

=15sq.units.= 15sq. units.
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