Question

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​

Answer 1

First integral

Since
`f(x)=3x+1` 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

`AC = 2,\ CB=6`

So, the area of the triangle is

`A_t = (2* 6)/(2)=6`

For the rectangle, we have

`A_r = AC* CE = 2* 4=8`

So, the total area is
`6+8=14`

Second integral

We have

`y = √(1-x^2) \implies y^2=1-x^2 \implies x^2+y^2=1`

So, this function is the upper half of the unit circle (the two halves are the functions
`y=\pm√(1-x^2)`)

Since the area of the unit circle is
`\pi`, the value of the integral is
`(\pi)/(2)`

Third integral

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