KCET · Maths · Area Under Curves
The area bounded by the curve \(y=\sin \left(\frac{x}{3}\right)\), \(x\)-axis and lines \(x=0\) and \(x=3 \pi\) is
- A 9
- B 0
- C 6
- D 3
Answer & Solution
Correct Answer
(C) 6
Step-by-step Solution
Detailed explanation
Required area \(=\int_{x=0}^{x=3 \pi} y d x=\int_{0}^{3 \pi} \sin \left(\frac{x}{3}\right) d x\)
Let \(\frac{x}{3}=t \Rightarrow d x=3 d t\)
Also, when
\(x=0\)
\(t=0\)
\(x=3 \pi\)
\(t=\pi\)
Then,
When
\(t=\pi\)
Then, required area
\(=\int_{0}^{\pi} \sin t(3 d t)=3[-\cos t]_{0}^{\pi}\)
\(=-3[\cos \pi-\cos 0]\)
\(=-3(-1-1)=-3(-2)=6\)
Let \(\frac{x}{3}=t \Rightarrow d x=3 d t\)
Also, when
\(x=0\)
\(t=0\)
\(x=3 \pi\)
\(t=\pi\)
Then,
When
\(t=\pi\)
Then, required area
\(=\int_{0}^{\pi} \sin t(3 d t)=3[-\cos t]_{0}^{\pi}\)
\(=-3[\cos \pi-\cos 0]\)
\(=-3(-1-1)=-3(-2)=6\)
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