Question

Graph the integrand and use areas to evaluate the integral

Answer 1

method 1
find the function whose derivative is
\[\frac{x}{2}+3\] then replace x by 4, replace x by -2 and subtract the second number from the first

Answer 2

method 2
graph the line
\[y=\frac{x}{2}+3\] and find the area of the figure beneath the line from x = -2 to x = 1. you will have a rectangle and a triangle so you can find the area using base time height for the rectangle and one half base times height for the triangle

Answer 3

sorry from x = -2 to x = 4

Answer 4

how do you get a rectangle from that graph?

Answer 5

I can see a triangle but not a rectangle?

Answer 6

you have on the graph (-2,2) and (4,5) yes?

Answer 7

yes

Answer 8

and draw a vertical line at x = -2 from the x axis to the point (-2,2) and another vertical line at x = 4 from the x - axis to the point (4,5)

Answer 9

oaky

Answer 10

you want the area of the figure bounded by the x - axis at the bottom, the line
\[y=\frac{x}{2}+3\] on top, the line
\[x=-2\] on the left and the line
\[x=4\] on the right

Answer 11

There is a triangle on the bottom and a triangle on top

Answer 12

i mean rectangle on bottom

Answer 13

yes!

Answer 14

rectangle has base 6 (from -2 to 4) and height 2 so area is 12

Answer 15

so we have to evuate -2 in order to get he height as well as evalute 4 to get the height

Answer 16

yes you know that if x = -2 then y = 2

Answer 17

and if x = 4 then y = 5

Answer 18

first one tells me height of rectangle is 2. base is 6 from the picture

Answer 19

now triangle also has base 6 but we have to be careful for the height

Answer 20

wait triangle has base 6?

Answer 21

looks like it to me. same as base of rectangle. from -2 to 4

Answer 22

okay, wow, that was tricky, cause i would have thought that the triangle had a base of 4

Answer 23

let me know if you see it

Answer 24

no triangle has base 6 . you are not starting at y axis you are starting at x = -2

Answer 25

and height of triangle is 3

Answer 26

right, so we find the area of those two figures and add them together right

Answer 27

do you see that the height is 3?

Answer 28

because rectangle has height 2 and the triangle sits above it and goes up to 5

Answer 29

Wait the height of the triangle is 3,??

Answer 30

yes. the base is the horizontal line y = 2 from (-2,2) to (4,2) so base is 6
the height is the vertical line from (4,2) to (4,5) so height it 3

Answer 31

if we are looking at the same picture it should be clear

Answer 32

so we get the 5 from evlauating the right endpoint, or x=4, but the height of the triangle is not 5, its really 3

Answer 33

and the base is still 6, even though the recatngel also has bsae 6

Answer 34

and eventhough the triangle sits on top of the rectangle

Answer 35

yes we can try here if it is not clear
http://www.twiddla.com/575798
although i am not that familiar with this