MHT CET · Maths · Area Under Curves
The area enclosed between the parabola \(y^2=4 x\) and the line \(y=2 x-4\) is
- A \(\frac{17}{3}\) sq. units
- B 15 sq. units
- C \(\frac{19}{3}\) sq. units
- D 9 sq. units
Answer & Solution
Correct Answer
(D) 9 sq. units
Step-by-step Solution
Detailed explanation
Putting \(x=\frac{y^2}{4}\) in \(y=2 x-4\), we get
\(\begin{aligned}
& y=2\left(\frac{y^2}{4}\right)-4 \\
& \Rightarrow y^2-2 y-8=0 \\
& \Rightarrow(y-4)(y+2)=0 \\
& \Rightarrow y=4,-2
\end{aligned}\)
\(\begin{aligned}
\therefore & \text { Required area }=\int_{-2}^4\left(\frac{y+4}{2}-\frac{y^2}{4}\right) \mathrm{d} y \\
& =\frac{1}{2}\left[\frac{y^2}{2}+4 y\right]_{-2}^4-\frac{1}{4}\left[\frac{y^3}{3}\right]_{-2}^4 \\
& =\frac{1}{2}[8+16-(2-8)]-\frac{1}{12}[64-(-8)] \\
& =15-6 \\
& =9 \text { sq. units }
\end{aligned}\)
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