Q 43 - Calculus

Question

Each of the integrals or integral expressions in Exercises $38-44$ represents the area of a region in the plane. Use polar coordinates to sketch the region and evaluate the expressions.

Step-by-Step Solution

Answer

The required graph is :-

Step 1. Given Information

We have given the following function :-

Step 2: Draw a diagram and identify given information

Draw a right triangle or appropriate diagram based on the problem description. Label the known angle and the given speed/distance. Identify which component (horizontal or vertical) corresponds to the quantity being asked for.

Step 3: Set up the trigonometric equation

Using the right triangle, the vertical component (altitude gain) is found using \(\text{vertical} = v \sin(\theta)\) and the horizontal component is found using \(\text{horizontal} = v \cos(\theta)\), where \(v\) is the speed and \(\theta\) is the angle of elevation.

Step 4: Substitute values and compute

Substitute the given numerical values into the trigonometric equation and compute the result. Make sure to use the correct units throughout the calculation.

Step 5: State the final answer

Express the final answer with appropriate units and in the context of the original problem.

Step 6: Conclude with the answer

The required graph is :-