Calculus

Application of Integrals

Easy

Based on the following given information answer the question

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

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

✅ Correct Option: 3

$y = x|x| = x^2$ for $x \geq 0$ (above x-axis) and $y = -x^2$ for $x < 0$ (below x-axis).

So the curve is in the third quadrant for $x < 0$ and first quadrant for $x > 0$ , passing through the origin.

The graph showing this odd-function shape with shading between $x = -1$ and $x = 1$ matches option 3.

Question 47

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

Question 48

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

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