Use a double integral to calculate the volume of the solid in the first octant bounded by the coordinate planes and the surface z =7-y-7x.
Angles in Circles
Angles within a circle are feasible to create with the help of different properties of the circle such as radii, tangents, and chords. The radius is the distance from the center of the circle to the circumference of the circle. A tangent is a line made perpendicular to the radius through its endpoint placed on the circle as well as the line drawn at right angles to a tangent across the point of contact when the circle passes through the center of the circle. The chord is a line segment with its endpoints on the circle. A secant line or secant is the infinite extension of the chord.
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A circular arc is the arc of a circle formed by two distinct points. It is a section or segment of the circumference of a circle. A straight line passing through the center connecting the two distinct ends of the arc is termed a semi-circular arc.
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![**Calculating Volume Using Double Integrals**
**Introduction:**
In this exercise, we will explore how to use a double integral to determine the volume of a solid located in the first octant. This solid is bounded by the coordinate planes and the surface defined by the equation:
\[ z = 7 - y - 7x^2 \]
**Objective:**
Our goal is to calculate the volume of this solid region above the \(xy\)-plane, within the specified bounds.
**Explanation:**
- **First Octant:** This refers to the region in a three-dimensional coordinate system where \(x\), \(y\), and \(z\) are all non-negative.
- **Coordinate Planes:** The planes defined by \(x=0\), \(y=0\), and \(z=0\).
**Procedure:**
1. **Identify the Bounds:**
- The region is where the surface intersects the coordinate planes.
2. **Setup the Double Integral:**
- Integrate over the region where \(z \geq 0\), which involves finding the limits of integration for \(x\) and \(y\) that satisfy the equation \(z = 7 - y - 7x^2 \geq 0\).
3. **Compute the Integral:**
- Using the appropriate integration techniques, evaluate the double integral to determine the volume.
By understanding these steps, you can apply similar methods to find volumes of other regions bounded by surfaces and planes using double integrals.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8735dee1-ec31-4346-b70c-e4d10f64e04c%2F52b194a3-9d37-4b5b-8160-5b8c48d67265%2Fflo0l4.png&w=3840&q=75)
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