ক্ষেত্রফল নির্ণয়

Area of the region bounded by y=∣x∣y=|x| and y=1−∣x∣y=1-|x| is-

হানি নাটস

For y=∣x∣    &    y=1−∣x∣ y=|x| ~~~~\&~~~~ y=1-|x|

∣x∣=1−∣x∣ |x|=1-|x|

⇒2∣x∣=1∴x=±12∴x=−12,12∴A=2∫012(1−∣x∣)−∣x∣)dx \begin{array}{l}\Rightarrow 2|x|=1 \quad \therefore x= \pm \frac{1}{2} \\ \therefore x=-\frac{1}{2}, \frac{1}{2} \\ \left.\therefore A=2 \int_{0}^{\frac{1}{2}}(1-|x|)-|x|\right) d x\end{array}

=2∫012(1−2∣x∣)dx=2[x−∣x∣2]012=2[−14]=∣−12∣ \begin{array}{l}=2 \int_{0}^{\frac{1}{2}}(1-2|x|) d x \\ =2\left[x-|x|^{2}\right]_{0}^{\frac{1}{2}} \\ =2\left[-\frac{1}{4}\right] \\ =|-\frac{1}{2}|\end{array}

= 12\displaystyle \frac{1}{2}

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