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Closed 4 years ago .Evaluate the integral by interpreting it in terms of areas. In other words, draw a picture of the region the integral represents, and find the area using high school geometry. $$ \int_0^4 |5x−2|dx $$54.6k 3 3 gold badges 37 37 silver badges 75 75 bronze badges asked Jun 30, 2020 at 21:54 21 1 1 silver badge 6 6 bronze badges $\begingroup$ What have you tried so far? $\endgroup$ Commented Jun 30, 2020 at 21:55
$\begingroup$ I tried finding ∫|5x−2|dx, a=0,b=0.4 + ∫|5x−2|dx, a=0.4,b=4 i got 32.8 but it was wrong ( i got the 0.4 by 5x-2>=0) $\endgroup$
Commented Jun 30, 2020 at 21:58$\begingroup$ Why don't you draw the plot as suggested and note the area you want is just 2 right triangles? Here is the suggestion from WolframAlpha $\endgroup$
Commented Jun 30, 2020 at 22:01$\begingroup$ @RouaSabbagh I see you are new to SO, so just remember to put what you tried into your question. This way, people know that you made a good faith effort and will be happy to help you $\endgroup$
Commented Jun 30, 2020 at 22:02So visually the absolute value means your function can be decomposed into two different lines over the region x: [0,4]. As you found the line changes direction at x = 0.4, as this is where |5x - 2| changes sign.
So if we let y = |5x - 2|, then at x = 0, y = 2; at x = 0.4, y = 0; at x = 4, y = 18. Draw a line between (0,2) and (0.4,0) and find the area over the region x:[0,0.4]. Then draw a line between (0,2) and (4,18), and find the area over this region.
Using the area for triangles, you should get $(1/2)(0.4)(2) + (1/2)(4-0.4)(18) = 32.8$
