Based on the following given information answer the question Consider the function f(x) = x, x, x R. The area of the region bounded by f(x) = x, x, x-axis and from x = -1 to x = 1 is : Option 1: 2/3 sq. units Option 2: 0 sq. units Option 3: 1/9 sq. units Option 4: 2/9 sq. units

Answer: 2/3 sq. units. Solution: Area = _(-1)^1, x, x, dx = _(-1)^1 x^2 dx (since, x, x, = x^2). = [ x^3/3 ]_(-1)^1 = 1/3 - (-1/3) = 2/3 sq. units.

Calculus

Application of Integrals

Medium

Based on the following given information answer the question

Consider the function $f(x) = x|x|$ , $x \in R$ .

The area of the region bounded by $f(x) = x|x|$ , x-axis and from $x = -1$ to $x = 1$ is :

$\frac{2}{3}$ sq. units

0 sq. units

$\frac{1}{9}$ sq. units

$\frac{2}{9}$ sq. units

✅ Correct Option: 1

Area $= \int_{-1}^{1} |x|x|| \, dx = \int_{-1}^{1} x^2 \, dx$ (since $|x|x|| = x^2$ ).

$= \left[ \frac{x^3}{3} \right]_{-1}^{1} = \frac{1}{3} - \left(-\frac{1}{3}\right) = \frac{2}{3}$ sq. units.

Question 46

The region bounded by curve $y = x|x|$ , x-axis and lines $x = \pm 1$ is best represented graphically by :

Question 47

The area of the region bounded by $f(x) = x|x|$ and x-axis from $x = 0$ to $x = 4$ is :

Question 49

The slope of the tangent drawn at the point whose x coordinates is 2 on the curve $y = x|x|$ .

Question 50

Consider the following statements for the curve $f(x) = x|x|$ and find that which of the following(s) are/is correct :

(A) $f(x) = x|x|$ is differentiable at $x = 0$ .

(B) $f(x) = x|x|$ is continuous at $x = 0$ but not differentiable at $x = 0$ .

(D) $f(x) = x|x|$ is symmetrical about y-axis.

Choose the correct answer from the options given below :