What is the value of the definite integral represented by a banana?

The image displays the numbers 1 and $-1$ at the top and bottom, respectively, and "$dx$" on the right. The banana itself represents the function being integrated. This setup visually represents a definite integral. The curved shape of the banana, combined with the limits from $-1$ to 1, is a common visual pun for the function $f(x) = \sqrt{1-x^2}$, which describes the upper half of a unit circle.

The integral represented is:

$$\int_{-1}^{1} \sqrt{1-x^2}\,dx$$

This integral represents the area under the curve $y = \sqrt{1-x^2}$ from $x=-1$ to $x=1$. Geometrically, this is the area of a semicircle with radius $r=1$.

The formula for the area of a semicircle is $\frac{1}{2}\pi r^2$.
Substitute $r=1$ into the formula:

$$\text{Area} = \frac{1}{2}\pi (1)^2$$ $$\text{Area} = \frac{1}{2}\pi (1)$$ $$\text{Area} = \frac{\pi}{2}$$

The value of the integral is:

$$\int_{-1}^{1} \sqrt{1-x^2}\,dx = \frac{\pi}{2}$$

The final answer is $\frac{\pi}{2}$.