Questions: Sketch the region of integration for the following integral. [ int0^pi / 4 int0^5 / cos (theta) f(r, theta) r d r d theta ] The region of integration is bounded by A. y=0, y=x, and x=5 B. y=0, y=x, and y=5 C. y=0, x=sqrt25-y^2, and y=5 D. y=0, y=sqrt25-x^2, and x=5 E. None of the above

Sketch the region of integration for the following integral. [ int0^pi / 4 int0^5 / cos (theta) f(r, theta) r d r d theta ] The region of integration is bounded by A. y=0, y=x, and x=5 B. y=0, y=x, and y=5 C. y=0, x=sqrt25-y^2, and y=5 D. y=0, y=sqrt25-x^2, and x=5 E. None of the above
Transcript text: Sketch the region of integration for the following integral. \[ \int_{0}^{\pi / 4} \int_{0}^{5 / \cos (\theta)} f(r, \theta) r d r d \theta \] The region of integration is bounded by A. $y=0, y=x$, and $x=5$ B. $y=0, y=x$, and $y=5$ C. $y=0, x=\sqrt{25-y^{2}}$, and $y=5$ D. $y=0, y=\sqrt{25-x^{2}}$, and $x=5$ E. None of the above
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Solution

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Solution Steps

Step 1: Identify the region of integration

The given integral is: \[ \int_{0}^{\pi / 4} \int_{0}^{5 / \cos (\theta)} f(r, \theta) r \, dr \, d\theta \]

The limits of integration for \( r \) are from \( 0 \) to \( \frac{5}{\cos(\theta)} \), which describes a line \( x = 5 \) in Cartesian coordinates. The limits for \( \theta \) are from \( 0 \) to \( \frac{\pi}{4} \), which corresponds to the line \( y = x \).

Step 2: Determine the correct boundary description

The region of integration is bounded by:

  • \( y = 0 \) (the x-axis),
  • \( y = x \) (the line at \( \theta = \frac{\pi}{4} \)),
  • \( x = 5 \) (the line described by \( r = \frac{5}{\cos(\theta)} \)).

Thus, the correct answer is option A: \( y=0, y=x \), and \( x=5 \).

Final Answer

The region of integration is bounded by \( y=0, y=x \), and \( x=5 \).

{"axisType": 3, "coordSystem": {"xmin": 0, "xmax": 6, "ymin": 0, "ymax": 6}, "commands": ["y = 0", "y = x", "x = 5"], "latex_expressions": ["$y = 0$", "$y = x$", "$x = 5$"]}

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