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Area Under the Curves question

2020 · 7 Jan · Shift 2 · Q36
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Area Under the Curves question

2020 · 7 Jan · Shift 2 · Q36

JEE MainMathematicsArea Under the CurvesMCQ+4 / −1
The area (in sq. units) of the region {(x, y) ∈\in∈ R2 | 4x2 ≤\le≤ y ≤\le≤ 8x + 12} is :
  1. A
    1253{{125} \over 3}3125​
  2. B
    1283{{128} \over 3}3128​
  3. C
    1273{{127} \over 3}3127​
  4. D
    1243{{124} \over 3}3124​
View written solutionFree

Correct answer: B

  1. Interpret the region

The region is given by

{(x,y)∈R2∣4x2≤y≤8x+12}.\{(x,y)\in \mathbb{R}^2 \mid 4x^2 \le y \le 8x+12\}.{(x,y)∈R2∣4x2≤y≤8x+12}.

So the area is the area between the curves:

  • lower curve: y=4x2y=4x^2y=4x2
  • upper curve: y=8x+12y=8x+12y=8x+12

We first find their points of intersection.

  1. Find intersection points

Set

4x2=8x+12.4x^2=8x+12.4x2=8x+12.

Then

4x2−8x−12=04x^2-8x-12=04x2−8x−12=0 x2−2x−3=0x^2-2x-3=0x2−2x−3=0 (x−3)(x+1)=0.(x-3)(x+1)=0.(x−3)(x+1)=0.

Hence,

x=−1,  3.x=-1,\;3.x=−1,3.

So the curves intersect at x=−1x=-1x=−1 and x=3x=3x=3.

  1. Set up the area integral

Between x=−1x=-1x=−1 and x=3x=3x=3, the line lies above the parabola. Therefore,

Area=∫−13[(8x+12)−4x2]dx.\text{Area}=\int_{-1}^{3}\big[(8x+12)-4x^2\big]dx.Area=∫−13​[(8x+12)−4x2]dx.
  1. Evaluate the integral
∫(8x+12−4x2)dx=4x2+12x−4x33.\int (8x+12-4x^2)dx = 4x^2+12x-\frac{4x^3}{3}.∫(8x+12−4x2)dx=4x2+12x−34x3​.

Thus,

Area=[4x2+12x−4x33]−13.\text{Area}=\left[4x^2+12x-\frac{4x^3}{3}\right]_{-1}^{3}.Area=[4x2+12x−34x3​]−13​.

At x=3x=3x=3:

4(9)+12(3)−4(27)3=36+36−36=36.4(9)+12(3)-\frac{4(27)}{3}=36+36-36=36.4(9)+12(3)−34(27)​=36+36−36=36.

At x=−1x=-1x=−1:

4(1)+12(−1)−4(−1)33=4−12+43=−8+43=−203.4(1)+12(-1)-\frac{4(-1)^3}{3}=4-12+\frac{4}{3}=-8+\frac{4}{3}=-\frac{20}{3}.4(1)+12(−1)−34(−1)3​=4−12+34​=−8+34​=−320​.

Therefore,

Area=36−(−203)=1083+203=1283.\text{Area}=36-\left(-\frac{20}{3}\right)=\frac{108}{3}+\frac{20}{3}=\frac{128}{3}.Area=36−(−320​)=3108​+320​=3128​.
  1. Check options

The correct option is

1283\boxed{\frac{128}{3}}3128​​

which is Option B.

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