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

$\frac{64}{3}$ sq. units

$\frac{16}{3}$ sq. units

16 sq. units

64 sq. units

✅ Correct Option: 1

For $x \geq 0$ , $x|x| = x^2$ .

Area $= \int_{0}^{4} x^2 \, dx = \left[ \frac{x^3}{3} \right]_{0}^{4} = \frac{64}{3}$ 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 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 area of the region bounded by f(x)=x∣x∣f(x) = x|x|f(x)=x∣x∣ and x-axis from x=0x = 0x=0 to x=4x = 4x=4 is :