Question

question 1 of 7
compute the area a bounded by the circle ( r = 5 ) and the rays ( \theta=\frac{pi}{2} ), and ( \theta=pi ) as an integral in polar coordinates.
(use symbolic notation and fractions where needed.)

Explanation:

Step1: Recall the formula for the area in polar coordinates

The formula for the area \(A\) in polar coordinates is \(A=\frac{1}{2}\int_{\alpha}^{\beta}r^{2}d\theta\), where \(r\) is the polar equation of the curve and \(\alpha\) and \(\beta\) are the limits of integration for the angle \(\theta\).

Step2: Identify the values of \(r\), \(\alpha\), and \(\beta\)

Given \(r = 5\), \(\alpha=\frac{\pi}{2}\), and \(\beta=\pi\).

Step3: Substitute the values into the formula

Substitute \(r = 5\), \(\alpha=\frac{\pi}{2}\), and \(\beta=\pi\) into the area formula:

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Step4: Evaluate the integral

Integrate \(\int_{\frac{\pi}{2}}^{\pi}d\theta=\theta\big|_{\frac{\pi}{2}}^{\pi}=\pi-\frac{\pi}{2}=\frac{\pi}{2}\)
Then \(A = \frac{25}{2}\times\frac{\pi}{2}=\frac{25\pi}{4}\)

Answer:

\(\frac{25\pi}{4}\)