compute the area enclosed by the graphs of y = x + 6 , y = x^3 , and y = -x/2

Question

amistre64 · Answer

whats your graph of the bounded region ...

anonymous · Answer

sketched it out, and looks like left bounds are the intersection of the linear functions, and right bounds are the intersection of x + x and x^3

anonymous · Answer

so, -4 to 2

anonymous · Answer

just as an aside, how would you solve x + 6 = x^3

amistre64 · Answer

complete the cube maybe, but i just wolfed it

anonymous · Answer

or it's obvious that it's two

amistre64 · Answer

wasnt obvious to me till now lol

anonymous · Answer

or to me until I saw the wolf graph of the three

amistre64 · Answer

ok, now is our boundary consistent, unchanging, along our interval?

anonymous · Answer

no, it changes at 0

amistre64 · Answer

then what are your thoughts as to how we work this?

anonymous · Answer

calc from -4 to 0, add it to from 0 to 2

amistre64 · Answer

very good, so whats our setup, what do you need help with

anonymous · Answer

reassurance, will post the final answer, these are much easier than some of the antiderivatives from previous chapters

anonymous · Answer

got 24 for A1

amistre64 · Answer

$$\int_{-4}^{0}x+6+\frac12x~dx+\int_{0}^{2}x+6-x^3~dx$$
$$\left.\frac12x^2+6x+\frac14x^2\right|_{-4}^{0}~+~\left.
\frac12x^2+6x-\frac14x^4\right|_0^{2}$$
$$\left.-\frac12(-4)^2-6(-4)-\frac14(-4)^2\right.~+~\left.
\frac12(2)^2+6(2)-\frac14(2)^4\right.$$

amistre64 · Answer

-8+24-4 = 12   for A1

anonymous · Answer

-3/4 * -64 - 24 = A1 ...is the area formula I cam up with after plugging in

amistre64 · Answer

show me your antiderivative

anonymous · Answer

oops, cubed -4, should be squared

anonymous · Answer

got the same as you, (3/4)x^2 + 6x

amistre64 · Answer

good :)

anonymous · Answer

0 - (3/4(-4)^2 - 24 = ...

amistre64 · Answer

-6(-4) = +24

anonymous · Answer

oops, darn parenthesis

anonymous · Answer

12

amistre64 · Answer

[3/4 (0)^2 + 6(0)] - [3/4 (-4)^2 +6(-4)]

[0 + 0] - [12 -24]

[0] - [-12] = 12

anonymous · Answer

A2 = 10

amistre64 · Answer

correct

anonymous · Answer

AA = 22

amistre64 · Answer

very good

anonymous · Answer

woot, getting funner

amistre64 · Answer

yeah, these are to build your confidence, theyll tear it down later ;)

anonymous · Answer

not me, probably done after this...bigger and better things to learn that computers can't do for us

anonymous · Answer

but yeah, this should be rudamentary for someone pursuing any sciences