# Solve $\int_{-2}^{2}|1-x^{2}\lvert.\text{d}x$ [IIT 1989; BIT Mesra 1996; Kurukshetra CEE 1998; MP PET 2002; Kerala (Engg.) 2002]

(A) 2

(B) 4

(C) 6

(D) 8

Answer

Verified

87.6k+ views

**Hint:**Integration means adding smaller functions to create a larger one. It is the inverse of differentiation, so it is also known as anti differentiation. Integrals with no integration limit are known as indefinite integrals. Integrals with an upper and lower limit are said to be definite integrals.

**Complete step by step solution:**The given integral is I = $\int_{-2}^{2}|1-x^{2}\lvert.\text{d}x$

Here the given function is f(x) = $\|1-x^{2}\lvert$ and the total range over which the integral is to be done is -2 to 2, with a lower limit of -2 and an upper limit of 2. We observe that the value of this function is negative in the range of -2 to -1, positive in the range of -1 to 1, and again negative in the range of 1 to 2.

So, the above integral can be written as

I = $-\int_{-2}^{-1}(1-x^{2}).\text{d}x+\int_{-1}^{1}(1-x^{2}).\text{d}x-\int_{1}^{2}(1-x^{2}).\text{d}x$

I = $-\left[x-\dfrac{x^3}{3}\right]_{-2}^{-1}+\left[x-\dfrac{x^3}{3}\right]_{-1}^{1}-\left[x-\dfrac{x^3}{3}\right]_{1}^{2}$

I = $-((-1)-\dfrac{-1^3}{3}-((-2)-\dfrac{-2^3}{3}))+((1)-\dfrac{1^3}{3}-((-1)-\dfrac{-1^3}{3})+((2)-\dfrac{2^3}{3}-((1)-\dfrac{1^3}{3}))$

I = $-(-1+\dfrac{1}{3}+2-\dfrac{8}{3})+(1-\dfrac{1}{3}+1-\dfrac{1}{3})-(2-\dfrac{8}{3}-1+\dfrac{1}{3})$

I = $\dfrac{4}{3}+\dfrac{4}{3}+\dfrac{4}{3}$

I = 4

Hence, the integration of $\|1-x^{2}\lvert$over the range of -2 to 2 is 4.

**Option ‘B’ is correct**

**Note:**The integration should be done carefully. Integration can be applied in daily life. It is used in chemistry to study radioactive decay reactions. It can be used to calculate the velocity and trajectory of the object. It is also used to calculate the centre of mass, centre of gravity, mass, and momentum of the satellites.

Last updated date: 30th Sep 2023

•

Total views: 87.6k

•

Views today: 1.87k

Recently Updated Pages

What is the Full Form of DNA and RNA

What are the Difference Between Acute and Chronic Disease

Difference Between Communicable and Non-Communicable

What is Nutrition Explain Diff Type of Nutrition ?

What is the Function of Digestive Enzymes

What is the Full Form of 1.DPT 2.DDT 3.BCG

Trending doubts

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

What is the menstrual cycle Which hormone regulates class 12 biology CBSE

The forelimbs of man cat bat and whale are A Analogous class 12 biology CBSE

The usual shape of growth curve is a Sigmoid b Zigzag class 12 biology CBSE

How do you define least count for Vernier Calipers class 12 physics CBSE

What is the Full Form of PVC, PET, HDPE, LDPE, PP and PS ?

Differentiate between insitu conservation and exsitu class 12 biology CBSE

Give 10 examples of unisexual and bisexual flowers

How much time does it take to bleed after eating p class 12 biology CBSE