The area bounded by the curves \(y=|x|-1\) and \(y=-|x|+1\) is
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The curve \(y=|x|-1\) is an upward-opening V with vertex \((0,-1)\); the curve \(y=-|x|+1\) is a downward-opening V with vertex \((0,1)\).
They meet where \(|x|-1=-|x|+1\Rightarrow 2|x|=2\Rightarrow |x|=1\), giving intersection points \((1,0)\) and \((-1,0)\), with \(y=0\) there.
The bounded region is a quadrilateral with vertices \((0,1),(1,0),(0,-1),(-1,0)\) — a rhombus (square rotated by 45°) whose diagonals lie along the axes.
The diagonal along the y-axis has length \(1-(-1)=2\), and the diagonal along the x-axis has length \(1-(-1)=2\).
The area of a rhombus is half the product of its diagonals: \(\dfrac{2\times 2}{2}=2\).
Important Questions from Area
What is the area of the triangle ABC with sides a = 10cm and c = 4cm angle B = 30°?
For the equations \(ax + (a^2 + 1)y = 4\) and \(4x + ay = a^2\), which of the following statements is TRUE?
What is the area of the parallelogram?
Consider the following statements :
1. If in a triangle ABC, A = 2B and b = c, then it must be an obtuse-angled triangle.
2. There exists no triangle ABC with A = 40°, B = 65° and a/c = sin 40° cosec 15°Which of the above statements is/are correct?
Which one of the following is the area of a sector of a circle of radius 10 cm formed by an arc length of 15 cm?
