Question

The area bounded between the parabola $y^2=4x$ and the line$y=2x-4$ is equal to

• A. $\frac{17}{3}$ square units
• B. $\frac{19}{3}$ square units
• C. $9$ square units
• D. $15$ square units

Answer 1

The correct option is C

$9$ square units

Explanation for the correct option:

Find the area bounded between the curve:

We have been given a curve and a line

$y^2=4x$ and $y=2x-4$

Points of intersection are

$$\begin{gathered} y^2=4x \\ \Rightarrow {(2x-4)}^2=4x\left[\because y=2x-4\right] \\ \Rightarrow x^2-5x+4=0 \\ \Rightarrow (x-1)(x-4)=0 \\ \Rightarrow x=1,4 \end{gathered}$$

If $x=1$

$$\begin{gathered} \Rightarrow y=2\times 1-4 \\ =-2 \end{gathered}$$

If $x=4$

$$\begin{gathered} \Rightarrow y=2\times 4-4 \\ =8-4 \\ =4 \end{gathered}$$

Required area

$$\begin{gathered} ={\int }_{-2}^4\left(\frac{y+4}{2}\right)dy-{\int }_{-2}^4\frac{y^2}{4}dy \\ =\frac{1}{2}{\left[\frac{y^2}{2}+4y\right]}_{-2}^4-\frac{1}{4}{\left[\frac{y^3}{3}\right]}_{-2}^4 \\ =\frac{1}{2}[8+16-(2-8\left)\right]-\frac{1}{12}[64+8] \\ =\frac{1}{2}\left[30\right]-\frac{1}{12}\left(72\right) \\ =9squnit \end{gathered}$$

Hence, option (C) is the correct answer.