Question

The area bounded by the curve \(y=4x^2\), the \(x\)-axis, the line \(x=0\) and the line \(x=1\) is

• A. \(2\)
• B. \(\frac{2}{3}\)
• C. \(\frac{1}{3}\)
• D. \(\frac{4}{3}\)

Hint:

Area under a curve \(y=f(x)\) from \(x=a\) to \(x=b\) is \(\int_a^b f(x)\,dx\).

Answer 1

The Correct Option is D

We are given the curve: \[ y=4x^2. \] The area is bounded by: \[ x=0,\qquad x=1, \] and the \(x\)-axis. The area under the curve from \(x=0\) to \(x=1\) is given by: \[ A=\int_{0}^{1}y\,dx. \] Substitute \[ y=4x^2. \] So, \[ A=\int_{0}^{1}4x^2\,dx. \] Take the constant \(4\) outside: \[ A=4\int_{0}^{1}x^2\,dx. \] Now integrate: \[ \int x^2\,dx=\frac{x^3}{3}. \] Therefore, \[ A=4\left[\frac{x^3}{3}\right]_{0}^{1}. \] Substitute the limits: \[ A=\frac{4}{3}\left[1^3-0^3\right]. \] \[ A=\frac{4}{3}(1-0). \] \[ A=\frac{4}{3}. \] Hence, the required area is \[ \frac{4}{3}. \]