Q116 / 160 · TS EAMCET 2019 (03 May Shift 1) · Mathematics

$\int_0^{10}\left(5-\sqrt{10 x-x^2}\right) d x=$

  • A
    $50-25 \pi$
  • B
    $(100-25 \pi)$
  • Correct answer
    C
    $\frac{1}{2}(100-25 \pi)$
  • D
    $\frac{1}{4}(100-25 \pi)$

Solution

$$ \begin{aligned} & \text { Let } \\ & I=\int_0^{10} 5-\sqrt{\left.10 x-x^2\right)} d x, I=\int_0^{10}\left(5-\sqrt{(5)^2-(x-5)^2} d x\right. \\ & I=\left[5 x-\frac{x-5}{2} \sqrt{10 x-x^2}-\frac{25}{2} \sin ^{-1} \frac{x-5}{5}\right]_0^{10} \\ & I=\left(50-0-\frac{25}{2} \sin ^{-1} 1\right)-\left(0-0-\frac{25}{2} \sin ^{-1}(-1)\right) \\ & I=50-\frac{25}{2} \times \frac{\pi}{2}-\frac{25}{2} \times \frac{\pi}{2} \\ & I=50-\frac{25 \pi}{2} \Rightarrow I=\frac{1}{2}(100-25 \pi) \end{aligned} $$

Hence, the correct answer is option: 'C'.

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