Question

The area of the region bounded by the curves y = x² and y = √x is (in square units)

• A. \(\frac{2}{3}\)
• B. \(\frac{1}{3}\)
• C. \(\frac{1}{6}\)
• D. \(\frac{5}{6}\)
• E. 1

Answer 1

The Correct Option is B

We are asked to find the area of the region bounded by the curves \( y = x^2 \) and \( y = \sqrt{x} \). 

To find the area, we first need to determine the points of intersection of the two curves. Set \( x^2 = \sqrt{x} \):

\( x^2 = \sqrt{x} \) implies \( x^4 = x \), which simplifies to \( x(x^3 - 1) = 0 \). This gives the solutions \( x = 0 \) and \( x = 1 \).

Thus, the curves intersect at \( x = 0 \) and \( x = 1 \). We will compute the area between the curves from \( x = 0 \) to \( x = 1 \).

The area between the curves is given by the integral:

\( \text{Area} = \int_0^1 \left( \sqrt{x} - x^2 \right) \, dx \).

Now, compute the integral:

\( \int_0^1 \sqrt{x} \, dx = \int_0^1 x^{1/2} \, dx = \left[ \frac{2}{3} x^{3/2} \right]_0^1 = \frac{2}{3} \),

and

\( \int_0^1 x^2 \, dx = \left[ \frac{x^3}{3} \right]_0^1 = \frac{1}{3} \).

Thus, the area is:

\( \text{Area} = \frac{2}{3} - \frac{1}{3} = \frac{1}{3} \).

The correct answer is \( \frac{1}{3} \).