Question

Evaluate the definite integral by interpreting it in terms of areas. of integral from 7 to 3 (4x-16)dx
I will also write this out as an equation below

Answer 1

Evaluate the definite integral by interpreting it in terms of the areas.
\[\int\limits_{3}^{7}(4x-16)dx=?\]

Answer 2

I know that this will be a line and I am pretty sure that this will then give me two triangles that I need to find the area of and add them up to find the area.

Answer 3

draw a picture, you will get a triangle and a rectangle

Answer 4

ok i lied you get two triangles
one above the \(x\) axis has positive area, below negative
http://www.wolframalpha.com/input/?i=4x-16+domain+3..7

Answer 5

Since you really want areas, you can break up the problem into evaluating the NEGATIVE of the function from x = 3 to 4 and then evaluating the original function from x = 4 to 7.

Answer 6

Now, it is true that the first triangle is under the x-axis. The area is positive (the negative integrand) while the actual integral from 3 to 4 is negative. So, you have to ask yourself what it is you really want. If you want the definite integral evaluated, you will have to take the area of the second triangle (the one that is from 4 to 7) and subtract the area of the first triangle (the one from 3 to 4, that is under the x-axis). You would have to do this because the problem requires that you evaluate the definite integral by considering areas.

Note: If the problem were simply to get the definite integral (without requiring the use of areas), you could just evaluate the integral from 3 to 7. You would get the same answer as subtracting the area of triangle one (below the x-axis) from triangle two (above the x-axis.)