CBSE 2025 · Region 6 · Set 1 · Q33 · 5 marks
Draw a rough sketch for the curve $\displaystyle \mathrm{y}=2+|\mathrm{x}+1|$. Using integration, find the area of the region bounded by the curve $\displaystyle \mathrm{y}=2+|\mathrm{x}+1|, \mathrm{x}=-4, \mathrm{x}=3$ and $\displaystyle \mathrm{y}=0$.
Marking-scheme solution
Correct Graph:
\[\begin{aligned}
\text { Required area }= & \int_{-4}^{-1}(2+|\mathrm{x}+1|) \mathrm{dx}+\int_{-1}^{3}(2+|\mathrm{x}+1|) \mathrm{dx} \\
= & \int_{-4}^{-1}(1-\mathrm{x}) d \mathrm{x}+\int_{-1}^{3}(3+\mathrm{x}) d \mathrm{x} \\
& =-\left.\frac{(1-\mathrm{x})^{2}}{2}\right|_{-4} ^{-1}+\left.\frac{(3+\mathrm{x})^{2}}{2}\right|_{-1} ^{3} \\
& =\frac{21}{2}+16=\frac{53}{2}
\end{aligned}
\]
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