If the area of the larger portion bounded between the curves x^2+y^2=25 and y=|x-1| is 14(…

Question

If the area of the larger portion bounded between the curves $x^2+y^2=25$ and $y=|x-1|$ is $\dfrac14(b\pi+c)$, $b,c \in \mathbb{N}$, then $b+c$ is equal to \_\_\_\_\_\_\_\_.

Answer: 77

Accepted Answer

The line $y=|x-1|$ meets the circle at $x=4$ (right branch) and $x=-3$ (left branch).

$$A_{\text{smaller}} = 25\pi - \int_{-3}^4\sqrt{25-x^2}\,dx + \frac12(4)(4) + \frac12(3)(3)$$

Evaluating $\displaystyle\int_{-3}^4\sqrt{25-x^2}\,dx = \left[\frac{x}{2}\sqrt{25-x^2}+\frac{25}{2}\sin^{-1}\frac{x}{5}\right]_{-3}^4 = 12 + \frac{25}{2}\cdot\frac{\pi}{2}$

$$A_{\text{larger}} = 25\pi + \frac{25}{2} - 12 - \frac{25\pi}{4} = \frac{75\pi}{4}+\frac12 = \frac14(75\pi+2)$$

So $b=75$, $c=2$, and $b+c = 77$.

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