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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​

computing definite integral by appealing to geometric formulas Graph the following-example-1

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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.

computing definite integral by appealing to geometric formulas Graph the following-example-1
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