Question

Use a double integral to find the area of the region D.

The x y-coordinate plane is given. There is a curve that encloses a region.
The curve, labeled r = √theta, starts at the origin, goes counterclockwise and away from the origin, and ends on the positive x-axis.
The region, labeled D, is the area enclosed by the curve and the positive x-axis.

Answer 1

The area of the region D, enclosed by the curve
`\( r = √(\theta) \)` in polar coordinates, can be found using the double integral
`\(\int_(0)^(2\pi)\int_(0)^(√(\theta)) r \,dr\,d\theta\)`. The result is
`\((\pi)/(2)\)` square units.

Step-by-step explanation:

To find the area of the region D in polar coordinates, we use a double integral. The given curve is
`\( r = √(\theta) \)`, and the region D is enclosed by this curve and the positive x-axis. The double integral to calculate the area is
`\(\int_(0)^(2\pi)\int_(0)^(√(\theta)) r \,dr\,d\theta\)`. The inner integral accounts for the radial distance from the origin to the curve, and the outer integral integrates over the angle
`\( \theta \)` from 0 to
`\( 2\pi \)`.

Evaluating the double integral involves integrating with respect to r and
`\( \theta \)`, considering the limits specified. The calculations lead to the final result of
`\((\pi)/(2)\)` square units, representing the area of the region D.

In summary, the double integral is a powerful tool for finding areas in polar coordinates. The specific limits and integrand are chosen based on the given curve, and the integration process results in the desired area of the region D.

Answer 2

To calculate the area of the polar region D bounded by the curve r = √θ and the positive x-axis, set up a double integral in polar coordinates with r ranging from 0 to √θ and θ from 0 to 2π, then evaluate the integral to find the total area.

Step-by-step explanation:

To find the area of the region D enclosed by the curve r = √θ and the positive x-axis using a double integral, one must set up an integral in polar coordinates, since the given curve is naturally described in these terms. The region is swept out as θ ranges from 0 to 2π (a full revolution), and r is given as a function of θ. Thus, the double integral in polar coordinates to find the area is:

A = ∫∫_D r dr dθ

To establish the limits of integration, we need to recall that for r = √θ, when θ = 0, r also equals 0, and this increases as θ increases, terminating again at r = 0 when θ completes the cycle back to the positive x-axis (2π). Therefore, the limits for r are from 0 to √θ and for θ from 0 to 2π. The integral in precise terms is:

A = ∫^{2π}_0 ∫^{√θ}_0 r dr dθ

To calculate the area of region D, we evaluate the integral:

Perform the inner integral with respect to r, from 0 to √θ.

Next, perform the outer integral with respect to θ, from 0 to 2π.

The solution will give the total area of the region D.