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The area of the region bounded by the curves y = |x−1| and y = 3 - |x| is
- a)3 sq.units
- b)4 sq. units
- c)6 sq. units
- d)2 sq. units
Correct answer is option 'B'. Can you explain this answer?
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The area of the region bounded by the curves y =|x−1| and y = 3 ...
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The area of the region bounded by the curves y =|x−1| and y = 3 ...
Explanation:
The given curves are y = |x| and y = 3 - |x|.
To find the area of the region bounded by these curves, we need to find the points of intersection of the two curves.
Finding Points of Intersection:
1. For y = |x| and y = 3 - |x|, we have:
|x| = 3 - |x|
2. Solving the above equation, we get:
x = 1.5
3. So, the points of intersection are at x = -1.5 and x = 1.5.
Calculating Area:
1. The area can be divided into two parts: one above the x-axis and one below the x-axis.
2. Area above x-axis:
- For x in [-1.5, 1.5], the curve y = 3 - |x| is above y = |x|.
- Area = ∫(3 - |x| - |x|) dx from -1.5 to 1.5
= 4 sq. units
3. Area below x-axis:
- For x in [-1.5, 1.5], the curve y = |x| is above y = 3 - |x|.
- Area = ∫(|x| - (3 - |x|)) dx from -1.5 to 1.5
= 4 sq. units
4. Total area = Area above x-axis + Area below x-axis
= 4 + 4
= 8 sq. units
Therefore, the area of the region bounded by the curves y = |x| and y = 3 - |x| is 4 sq. units.
The given curves are y = |x| and y = 3 - |x|.
To find the area of the region bounded by these curves, we need to find the points of intersection of the two curves.
Finding Points of Intersection:
1. For y = |x| and y = 3 - |x|, we have:
|x| = 3 - |x|
2. Solving the above equation, we get:
x = 1.5
3. So, the points of intersection are at x = -1.5 and x = 1.5.
Calculating Area:
1. The area can be divided into two parts: one above the x-axis and one below the x-axis.
2. Area above x-axis:
- For x in [-1.5, 1.5], the curve y = 3 - |x| is above y = |x|.
- Area = ∫(3 - |x| - |x|) dx from -1.5 to 1.5
= 4 sq. units
3. Area below x-axis:
- For x in [-1.5, 1.5], the curve y = |x| is above y = 3 - |x|.
- Area = ∫(|x| - (3 - |x|)) dx from -1.5 to 1.5
= 4 sq. units
4. Total area = Area above x-axis + Area below x-axis
= 4 + 4
= 8 sq. units
Therefore, the area of the region bounded by the curves y = |x| and y = 3 - |x| is 4 sq. units.
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The area of the region bounded by the curves y =|x−1| and y = 3 ...
(a) 3 sq.unit
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The area of the region bounded by the curves y =|x−1| and y = 3 -|x| isa)3 sq.unitsb)4 sq. unitsc)6 sq. unitsd)2 sq. unitsCorrect answer is option 'B'. Can you explain this answer?
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