Computing definite integral by appealing to geometric formulas graph the following definite integral then find the exact area of the shaded region justify your answer for each problem

First integral

Since is a linear function, its graph is a line. This means that you can decompose the area under its graph as the sum of a right triangle and a rectangle, as shown in the attached image.

All the dimensions are fairly easy to deduce: we have

So, the area of the triangle is

For the rectangle, we have

So, the total area is

Second integral

We have

So, this function is the upper half of the unit circle (the two halves are the functions )

Since the area of the unit circle is , the value of the integral is

Third integral

It follows the exact same logic as the first one, you only have to adjust the numbers.