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.
Arcs in Circles
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.
![**Problem Statement:**
Find the arc length of the graph given below over the interval \([0, 7]\).
\[ y = \sqrt{49 - x^2} \]
**Graph and Diagram Explanation:**
This problem is asking for the calculation of the arc length of the function \( y = \sqrt{49 - x^2} \) over the specified interval of \( x \) values from 0 to 7.
The function represents the upper half of a circle with a radius of 7, centered at the origin (0, 0). The arc length from 0 to 7 will essentially measure part of the circumference of this semicircle.
To find the arc length \( L \) of a curve \( y = f(x) \) from \( x = a \) to \( x = b \), use the formula:
\[ L = \int_a^b \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx \]
In this case, determine \(\frac{dy}{dx}\), substitute into the formula, and evaluate the integral over the given interval.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe15d21cd-3875-4270-92b4-6749e1bc7456%2F5ca1acc0-1f8d-4022-84d4-6d4d99dffe2e%2Fte0cx0g_processed.png&w=3840&q=75)
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