Question

How many definite integrals would be required to represent the area of the region enclosed by the curves y = cos2x sinx and y = 0.03x2, assuming you could not use the absolute value function?

1
2
3
4
5

Find the area of the region bounded by the curve y = f(x) = x3 – 4x + 1 and the tangent line to the curve y = f(x) at (–1,4).

You get:








8.25

6.25

Answer 1

For the first one, look at the graph (attached). Since you can't use the absolute value functions, how many different regions are defined?

Answer 2

4?

Answer 3

Actually 5. There is a small area (I don't know if you can even see the shading) that you probably missed.
For the second one, first you must find the equation of the tangent line first, the integrate.

Answer 4

thnxa

Answer 5

does that mean find derivative

Answer 6

Yes, then evaluate it at the given x-value.

Answer 7

could u help me with some more plz

Answer 8

sure.

Answer 9

thnxs

Answer 10

It looks like you are dealing with average value. You can find this with:

\[\frac{1}{b-a}\int\limits_{a}^{b}f(x)dx\] I assume you can use a calculator?

Answer 11

yup

Answer 12

@BTaylor

Answer 13

Ok, so can you set up the integral for #6?

Answer 14

ok give me a sec and check my work when I post k? and thnxs

Answer 15

sure thing.

Answer 16

hold one

Answer 17

\[1/c-1\int\limits_{1}^{c}(9\pi/x^2)\cos(\pi/x)dx\]

Answer 18

is this right?

Answer 19

yes, but according to the problem it is set equal to -.9
Now, multiply both sides by (c-1). Using your calculator, find the indefinite integral.

Answer 20

k

Answer 21

give me a sec

Answer 22

\[(1/c-1)\int\limits_{1}^{c}(9\pi/x^2)\cos(\pi/x)=-0.9\]

Answer 23

I need some help @BTaylor

Answer 24

@sourwing could u help me plz

Answer 25

sorry, had to go for a sec. You will end up with
\[\int\limits_{1}^{c}\frac{9 \pi}{x^2}\cos \left( \frac{\pi}{x} \right)dx=-0.9(c-1)\]

Answer 26

then what?

Answer 27

take the antiderivative/indefinite integral of the left side. Using the fundamental theorem of calculus, set it up so you have F(c)-F(1) -> F(x) is the antiderivative of f(x) evaluated at x.

Answer 28

From there you should be able to solve.

Answer 29

could u show me how to do it on this one plz

Answer 30

I'll try. The indefinite integral is actually quite simple: \(F(x)=-9sin \left(\frac{\pi}{x} \right)+C \)

Answer 31

So, we get: \[-9\sin \left( \frac{\pi}{c} \right) +9\sin(1)=-.9(c-1)\]

Answer 32

Use your calculator to solve for c.

Answer 33

wait...looking at the question, just plug in for c the different multiple choice values. Saves you work to solve.
*facepalm*

Answer 34

great thnxs, could u help me with another one plzzz

Answer 35

sure? which one?

Answer 36

7

Answer 37

This one is slightly easier. You know you need to find the average value, and this time it gives you the bounds. Using the formula \[average ~ value = \frac{1}{b-a}\int\limits_{a}^{b}f(x)dx\]can you find the average value of that polynomial?

Answer 38

yah, but im still working on the other one so brb

Answer 39

is 6 e?@BTaylor

Answer 40

@BTaylor

Answer 41

is 6 e?

Answer 42

Sounds right to me.

Answer 43

cool ok now 7.

Answer 44

can u help me with 8 plz

Answer 45

@BTaylor

Answer 46

just use that average value formula and plug it into your calculator.

Answer 47

is that the answer?

Answer 48

yep. Since the question (for #8) asks for the average value, you just need to answer with the average value.