To evaluate the integral
[tex]\[
\int_{-7}^7 \sqrt{7^2-x^2} \, dx,
\][/tex]
we can interpret it in terms of geometric areas.
1. Recognize the Integrand: The expression [tex]\(\sqrt{7^2 - x^2}\)[/tex] resembles the equation for the upper half of a circle centered at the origin with radius [tex]\(7\)[/tex]. Specifically, the equation of the full circle is given by:
[tex]\[
x^2 + y^2 = 7^2 \implies y = \pm \sqrt{7^2 - x^2}
\][/tex]
The integral given, [tex]\(\sqrt{7^2 - x^2}\)[/tex], represents the upper semicircle of this circle.
2. Identify the Limits of Integration: The limits of integration are from [tex]\(-7\)[/tex] to [tex]\(7\)[/tex], which span the entire diameter of the circle along the [tex]\(x\)[/tex]-axis.
3. Geometric Interpretation: The integral
[tex]\[
\int_{-7}^7 \sqrt{7^2 - x^2} \, dx
\][/tex]
represents the area under the curve [tex]\(y = \sqrt{7^2 - x^2}\)[/tex] from [tex]\(x = -7\)[/tex] to [tex]\(x = 7\)[/tex]. This is the upper half of the circle, or more precisely, the area of a semicircle with radius 7.
4. Calculate the Area of the Semicircle:
- The formula for the area [tex]\(A\)[/tex] of a full circle with radius [tex]\(r\)[/tex] is:
[tex]\[
A_{\text{full circle}} = \pi r^2
\][/tex]
- Here, the radius [tex]\(r\)[/tex] is 7, so the area of the full circle is:
[tex]\[
A_{\text{full circle}} = \pi \cdot 7^2 = 49\pi
\][/tex]
- Since the integral only covers the upper half of the circle, we need the area of the semicircle, which is half the area of the full circle:
[tex]\[
A_{\text{semicircle}} = \frac{1}{2} \times 49\pi = \frac{49\pi}{2}
\][/tex]
5. Numerical Value:
- Using the known value of [tex]\(\pi \approx 3.141592653589793\)[/tex], the area of the full circle is approximately:
[tex]\[
49 \times 3.141592653589793 \approx 153.93804002589985
\][/tex]
- Therefore, the area of the semicircle is:
[tex]\[
\frac{153.93804002589985}{2} \approx 76.96902001294993
\][/tex]
So, the value of the integral is:
[tex]\[
\boxed{76.96902001294993}
\][/tex]
