Question

pleeeeeaseee help meeeee computing definite integral by appealing to geometric formulas

Graph the following integral then, find the exact area of the shaded region. Justify for each problem ​

Answer 1

Plotting the functions is highly recommended.

For A, the area under $f(x)=3$ over the interval [1, 2] is the area of rectangle with height 3 and width 2 - 1 = 1, so the area is

$\displaystyle\int_1^23\,\mathrm dx=3\cdot1=3$

For B, the area under $f(x)=1-|x-1|$ plotted over the interval [0, 2] is the area of a triangle with height 1 and base length 2, so the area is

$\displaystyle\int_0^2(1-|x-1|)\,\mathrm dx=\frac12\cdot1\cdot2=1$