MHT CET · Maths · Area Under Curves
The area (in \(s q\). units) bounded by the curve \(y=x|x|, \mathrm{X}\)-axis and the lines \(x=-1\) and \(x=1\) is
- A \(\frac{2}{3}\)
- B \(\frac{1}{3}\)
- C 1
- D \(\frac{4}{3}\)
Answer & Solution
Correct Answer
(A) \(\frac{2}{3}\)
Step-by-step Solution
Detailed explanation
\(y=x|x| ...[Given]\)
Required area
\(\begin{aligned}
& =\int_{-1}^1 x|x| \mathrm{d} x \\
& =2 \int_0^1 x^2 \mathrm{~d} x \quad \ldots[\because \text { Area is always positive }] \\
& =2 \times\left[\frac{x^3}{3}\right]_0^1 \\
& =2 \times\left(\frac{1}{3}-0\right)=\frac{2}{3} \text { sq.units }
\end{aligned}\)
Required area
\(\begin{aligned}
& =\int_{-1}^1 x|x| \mathrm{d} x \\
& =2 \int_0^1 x^2 \mathrm{~d} x \quad \ldots[\because \text { Area is always positive }] \\
& =2 \times\left[\frac{x^3}{3}\right]_0^1 \\
& =2 \times\left(\frac{1}{3}-0\right)=\frac{2}{3} \text { sq.units }
\end{aligned}\)
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