Q87 / 180 · KCET 2015 · Mathematics
Area bounded by \( y=x^{3}, y=8 \) and \( x=0 \) is
- A\( 2 \) sq. units
- B\( 14 \mathrm{sq} \). units
- Correct answerC\( 12 \mathrm{sq} \). units
- D\( 6 \) sq. units
Solution
Given equation,
$y=x^{3} \rightarrow(1)$ $y=8 \rightarrow(2)$ $x=0 \rightarrow(3)$
Now, \( x=y^{1 / 3} \)
So, required area is given by \( \int_{0}^{8} x d y \)
\[ \begin{array}{l} =\int_{0}^{8} y^{1 / 3} d y=\left[\frac{3}{4} y^{4 / 3}\right]_{0}^{8}=\frac{3}{4}\left(8^{4 / 3}\right) \\ =\frac{3}{4}\left(2^{4}\right)=12 \end{array} \]
$y=x^{3} \rightarrow(1)$ $y=8 \rightarrow(2)$ $x=0 \rightarrow(3)$
Now, \( x=y^{1 / 3} \)
So, required area is given by \( \int_{0}^{8} x d y \)
\[ \begin{array}{l} =\int_{0}^{8} y^{1 / 3} d y=\left[\frac{3}{4} y^{4 / 3}\right]_{0}^{8}=\frac{3}{4}\left(8^{4 / 3}\right) \\ =\frac{3}{4}\left(2^{4}\right)=12 \end{array} \]
Hence, the correct answer is option: 'C'.