KCET · Maths · Area Under Curves
The area of the region bounded by the line \(y=3 x\) and the curve \(y=x^2\) sq units is
- A \(10\)
- B \(9 / 2\)
- C \(9\)
- D \(5\)
Answer & Solution
Correct Answer
(B) \(9 / 2\)
Step-by-step Solution
Detailed explanation
We have, \(y=3 x\) and \(y=x^2 \Rightarrow 3 x=x^2\) \(x=3\) and \(x=0\) are the points of intersection.
Now, area \(=\int_0^3\left(3 x-x^2\right) d x\)
\(=\left.\frac{3 x^2}{2}\right|_0 ^3-\left.\frac{x^3}{3}\right|_0 ^3\)
\(=3 \cdot \frac{9}{2}-\frac{27}{3}\)
\(=\frac{27}{2}-9=\frac{9}{2}\) sq units
Now, area \(=\int_0^3\left(3 x-x^2\right) d x\)
\(=\left.\frac{3 x^2}{2}\right|_0 ^3-\left.\frac{x^3}{3}\right|_0 ^3\)
\(=3 \cdot \frac{9}{2}-\frac{27}{3}\)
\(=\frac{27}{2}-9=\frac{9}{2}\) sq units
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