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Area Under the Curves question

2019 · 12 Jan · Shift 1 · Q40
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  5. /2019 · 12 Jan · Shift 1 · Q40

Area Under the Curves question

2019 · 12 Jan · Shift 1 · Q40

JEE MainMathematicsArea Under the CurvesMCQ+4 / −1
The area (in sq. units) of the region bounded by the parabola, y = x2 + 2 and the lines, y = x + 1, x = 0 and x = 3, is
  1. A
    154{{15} \over 4}415​
  2. B
    152{{15} \over 2}215​
  3. C
    212{{21} \over 2}221​
  4. D
    174{{17} \over 4}417​
View written solutionFree

Correct answer: B

  1. Given curves and boundaries

We need the area bounded by:

  • Parabola: y=x2+2y = x^2 + 2y=x2+2
  • Line: y=x+1y = x + 1y=x+1
  • Vertical lines: x=0x=0x=0 and x=3x=3x=3

So the required area is the area between the two curves from x=0x=0x=0 to x=3x=3x=3.

  1. Find which curve is above the other

Compare: y1=x2+2,y2=x+1y_1 = x^2+2, \qquad y_2 = x+1y1​=x2+2,y2​=x+1

Compute: y1−y2=(x2+2)−(x+1)=x2−x+1y_1-y_2 = (x^2+2)-(x+1)=x^2-x+1y1​−y2​=(x2+2)−(x+1)=x2−x+1

Now, x2−x+1=(x−12)2+34>0x^2-x+1 = \left(x-\frac12\right)^2+\frac34 > 0x2−x+1=(x−21​)2+43​>0 for all real xxx.

Hence, x2+2>x+1x^2+2 > x+1x2+2>x+1 for every xxx. Therefore, on [0,3][0,3][0,3], the parabola lies above the line.

  1. Set up the area integral

Thus area is A=∫03[(x2+2)−(x+1)] dxA = \int_0^3 \Big[(x^2+2)-(x+1)\Big] \, dxA=∫03​[(x2+2)−(x+1)]dx A=∫03(x2−x+1) dxA = \int_0^3 (x^2-x+1)\,dxA=∫03​(x2−x+1)dx

  1. Evaluate the integral

∫(x2−x+1) dx=x33−x22+x\int (x^2-x+1)\,dx = \frac{x^3}{3}-\frac{x^2}{2}+x∫(x2−x+1)dx=3x3​−2x2​+x

So, A=[x33−x22+x]03A = \left[\frac{x^3}{3}-\frac{x^2}{2}+x\right]_0^3A=[3x3​−2x2​+x]03​

At x=3x=3x=3: 273−92+3=9−92+3=12−92=24−92=152\frac{27}{3}-\frac{9}{2}+3 = 9-\frac92+3 = 12-\frac92 = \frac{24-9}{2} = \frac{15}{2}327​−29​+3=9−29​+3=12−29​=224−9​=215​

At x=0x=0x=0: 000

Therefore, A=152A = \frac{15}{2}A=215​

  1. Check options
  • A: 154\frac{15}{4}415​
  • B: 152\frac{15}{2}215​
  • C: 212\frac{21}{2}221​
  • D: 174\frac{17}{4}417​

So the correct option is: B 152\boxed{\text{B } \frac{15}{2}}B 215​​

  1. Comparison with stored answer

Stored correct answer: B

Our derived answer is also B. So they agree.

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