Question

In Exercises 9, a graph of a function f(x) is given. Using the geometry of the graph, evaluate the definite integrals?

Answer 1

The integral equals `pi`.

Explanation:

We know that the integral from `a` to `b` of a certain function `f(x)` represents the area of the curve between `a` and `b` and between the function and the x-axis.

We see that the graph is a half circle with radius `2`, thus the area would be `(2^2pi)/2 = 2pi` square units. If you draw a vertical line at `x = 2`, you will see the half circle is divided into two equal parts.

Thus, `int_0^2 f(x) dx = (int_0^4 f(x) dx)/2`

`int_0^2 f(x) dx = (2pi)/2`

`int_0^2 f(x) dx = pi`

Hopefully this helps!