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Area Under the Curves question

2019 · 9 Jan · Shift 2 · Q40
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  5. /2019 · 9 Jan · Shift 2 · Q40

Area Under the Curves question

2019 · 9 Jan · Shift 2 · Q40

JEE MainMathematicsArea Under the CurvesMCQ+4 / −1
The area of the region A = {(x, y) : 0 ≤\le≤ y ≤\le≤ x |x| + 1 and −-− 1 ≤\le≤ x ≤\le≤ 1} in sq. units, is :
  1. A
    23{2 \over 3}32​
  2. B
    2
  3. C
    43{4 \over 3}34​
  4. D
    13{1 \over 3}31​
View written solutionFree

Correct answer: B

  1. Interpret the region

The set is

A={(x,y):0≤y≤x∣x∣+1, −1≤x≤1}.A=\{(x,y): 0\le y\le x|x|+1,\,-1\le x\le 1\}.A={(x,y):0≤y≤x∣x∣+1,−1≤x≤1}.

So for each x∈[−1,1]x\in[-1,1]x∈[−1,1], the vertical strip goes from y=0y=0y=0 to y=x∣x∣+1.y=x|x|+1.y=x∣x∣+1.

Hence the required area is

Area=∫−11(x∣x∣+1) dx.\text{Area}=\int_{-1}^{1} (x|x|+1)\,dx.Area=∫−11​(x∣x∣+1)dx.
  1. Write x∣x∣x|x|x∣x∣ piecewise

Recall:

  • If x≥0x\ge 0x≥0, then ∣x∣=x|x|=x∣x∣=x, so x∣x∣=x2x|x|=x^2x∣x∣=x2.
  • If x<0x<0x<0, then ∣x∣=−x|x|=-x∣x∣=−x, so x∣x∣=−x2x|x|=-x^2x∣x∣=−x2.

Thus,

x∣x∣={−x2,−1≤x<0,x2,0≤x≤1.x|x|= \begin{cases} -x^2, & -1\le x<0,\\[4pt] x^2, & 0\le x\le 1. \end{cases}x∣x∣={−x2,x2,​−1≤x<0,0≤x≤1.​

Therefore,

Area=∫−10(1−x2) dx+∫01(1+x2) dx.\text{Area}=\int_{-1}^{0} (1-x^2)\,dx+\int_{0}^{1}(1+x^2)\,dx.Area=∫−10​(1−x2)dx+∫01​(1+x2)dx.
  1. Evaluate the first integral
∫−10(1−x2) dx=[x−x33]−10.\int_{-1}^{0}(1-x^2)\,dx =\left[x-\frac{x^3}{3}\right]_{-1}^{0}.∫−10​(1−x2)dx=[x−3x3​]−10​.

At x=0x=0x=0:

0−0=0.0-0=0.0−0=0.

At x=−1x=-1x=−1:

−1−(−1)33=−1+13=−23.-1-\frac{(-1)^3}{3}=-1+\frac13=-\frac23.−1−3(−1)3​=−1+31​=−32​.

So,

∫−10(1−x2) dx=0−(−23)=23.\int_{-1}^{0}(1-x^2)\,dx=0-\left(-\frac23\right)=\frac23.∫−10​(1−x2)dx=0−(−32​)=32​.
  1. Evaluate the second integral
\int_{0}^{1}(1+x^2)\,dx =\left[x+\frac{x^3}{3}\right]_{0}^{1} =1+\frac13= rac43.
  1. Add the two parts
Area=23+43=63=2.\text{Area}=\frac23+\frac43=\frac63=2.Area=32​+34​=36​=2.
  1. Match with options

The area is

2.2.2.

So the correct option is B.

  1. Comparison with stored answer

Stored correct answer: B

Our derived answer: B

They agree.

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