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

$A_t = \dfrac{2\times 6}{2}=6$

For the rectangle, we have

$A_r = AC\times CE = 2\times 4=8$

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

Second integral

We have

$y = \sqrt{1-x^2} \implies y^2=1-x^2 \implies x^2+y^2=1$

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

Since the area of the unit circle is $\pi$, the value of the integral is $\frac{\pi}{2}$

Third integral

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