91Ó°ÊÓ

A double integral involves integrating over a specific region within a plane. In this case, the double integral is expressed as \( \int_{0}^{4} \int_{-\sqrt{16-y^{2}}}^{\sqrt{16-y^{2}}} \sqrt{25-x^{2}-y^{2}} \, dx \, dy \).
Here, the limits for \( x \) suggest that for each specific value of \( y \), \( x \) ranges from \( -\sqrt{16-y^{2}} \) to \( \sqrt{16-y^{2}} \), forming a semicircle. The upper and lower limits of the outer integral with respect to \( y \), from 0 to 4, indicate the vertical section within which this semicircle lies.
In simpler terms, as \( y \) varies from 0 to 4, we gather all \( x \) values that can form a semicircle, which reflects a part of the circle \( x^2 + y^2 = 16 \). This means the region of integration stretches over the

• Upper half of a circle with radius 4
• Symmetrical across the x-axis with \( y \) values limited from 0 to 4

The function we are integrating is \( \sqrt{25-x^{2}-y^{2}} \), representing the upper hemisphere of a sphere with a complete equation \( x^2 + y^2 + z^2 = 25 \).
This function, as a result, forms a spherical cap—a portion of a sphere cut off by a plane.
At any chosen point \((x, y)\) in the defined region, this tells us the height \( z \) extends up to \( \sqrt{25-x^2-y^2} \). This correlation yields the familiar shape of a dome capped from the top of the sphere.
The spherical cap thus:

• Is built over the semicircle identified in the \( xy \)-plane.
• Rises to a maximum height of 5 units when \( x=0 \) and \( y=0 \), reaching the top point of this hemisphere.

The semicircle forms the base of our spherical cap in the integration process. To visualize this part, think of a circle \( x^2 + y^2 = 16 \), which is cut in half along the x-axis.
Even though the equation delineates a full circle, by setting the restrictions \( y \) only between 0 and 4, we effectively consider just the upper hemisphere.
This semicircle:

• Has a radius of 4—that’s because \( x^2 + y^2 \leq 16 \) outlines a full circle with this radius.
• Occupies space only where \( y \) is positive from 0 to 4.

Hence, the semicircle here serves as the floor inside the circle \( x^2 + y^2 = 16 \), underlying the spherical cap that emerges with the integral.

Volume calculation in the context of a double integral involves assessing the space under a surface across the region of integration. The double integral \( \int_{0}^{4} \int_{-\sqrt{16-y^{2}}}^{\sqrt{16-y^{2}}} \sqrt{25-x^{2}-y^{2}} \, dx \, dy \) essentially computes the volume of the spherical cap.
This is achieved by evaluating the sum of all infinitesimally small volumes \( \sqrt{25-x^2-y^2} \, dx \, dy \) over the integration region. Since these define the height as \( z \) above every point \( (x, y) \), the computation extends upward within the hemisphere.
When implemented:

• The inner integral calculates the width along the x-axis for each slice of y.
• The outer integral sums up these slices as y moves from 0 to 4.

Ultimately, solving this integral determines the precise volume contained within the base semicircle and the curved cap, representing the spatial content over this region.